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Theorem f1oweALT 7978
Description: Alternate proof of f1owe 7357, more direct since not using the isomorphism predicate, but requiring ax-un 7738. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
f1oweALT.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
Assertion
Ref Expression
f1oweALT (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem f1oweALT
Dummy variables 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ofo 6842 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 df-fo 6552 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
3 freq2 5645 . . . . . . 7 (ran 𝐹 = 𝐵 → (𝑆 Fr ran 𝐹𝑆 Fr 𝐵))
43biimprd 247 . . . . . 6 (ran 𝐹 = 𝐵 → (𝑆 Fr 𝐵𝑆 Fr ran 𝐹))
5 df-fn 6549 . . . . . . 7 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6 df-fr 5629 . . . . . . . . . . . . . . . . . . . 20 (𝑆 Fr ran 𝐹 ↔ ∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢))
7 vex 3466 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
87funimaex 6639 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝐹 → (𝐹𝑧) ∈ V)
9 n0 4346 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ≠ ∅ ↔ ∃𝑤 𝑤𝑧)
10 funfvima2 7240 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑤𝑧 → (𝐹𝑤) ∈ (𝐹𝑧)))
11 ne0i 4334 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹𝑤) ∈ (𝐹𝑧) → (𝐹𝑧) ≠ ∅)
1210, 11syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑤𝑧 → (𝐹𝑧) ≠ ∅))
1312exlimdv 1929 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (∃𝑤 𝑤𝑧 → (𝐹𝑧) ≠ ∅))
149, 13biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝐹𝑧) ≠ ∅))
1514imp 405 . . . . . . . . . . . . . . . . . . . . . . 23 (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝐹𝑧) ≠ ∅)
16 imassrn 6072 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑧) ⊆ ran 𝐹
1715, 16jctil 518 . . . . . . . . . . . . . . . . . . . . . 22 (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅))
18 sseq1 4004 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (𝑤 ⊆ ran 𝐹 ↔ (𝐹𝑧) ⊆ ran 𝐹))
19 neeq1 2993 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (𝑤 ≠ ∅ ↔ (𝐹𝑧) ≠ ∅))
2018, 19anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝐹𝑧) → ((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) ↔ ((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅)))
21 raleq 3312 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (∀𝑓𝑤 ¬ 𝑓𝑆𝑢 ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
2221rexeqbi1dv 3324 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝐹𝑧) → (∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
2320, 22imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝐹𝑧) → (((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) ↔ (((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2423spcgv 3581 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2517, 24syl7 74 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
268, 25syl 17 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝐹 → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
276, 26biimtrid 241 . . . . . . . . . . . . . . . . . . 19 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2827com23 86 . . . . . . . . . . . . . . . . . 18 (Fun 𝐹 → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2928expd 414 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))))
3029anabsi5 667 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
3130impd 409 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
32 fores 6817 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝐹𝑧):𝑧onto→(𝐹𝑧))
33 vex 3466 . . . . . . . . . . . . . . . . . . . . . 22 𝑣 ∈ V
34 vex 3466 . . . . . . . . . . . . . . . . . . . . . 22 𝑤 ∈ V
35 fveq2 6893 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
3635breq1d 5155 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑣 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑣)𝑆(𝐹𝑦)))
37 fveq2 6893 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
3837breq2d 5157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → ((𝐹𝑣)𝑆(𝐹𝑦) ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
39 f1oweALT.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
4033, 34, 36, 38, 39brab 5541 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑅𝑤 ↔ (𝐹𝑣)𝑆(𝐹𝑤))
41 fvres 6912 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑧 → ((𝐹𝑧)‘𝑣) = (𝐹𝑣))
42 fvres 6912 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝑧 → ((𝐹𝑧)‘𝑤) = (𝐹𝑤))
4341, 42breqan12rd 5162 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑧𝑣𝑧) → (((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
4440, 43bitr4id 289 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑧𝑣𝑧) → (𝑣𝑅𝑤 ↔ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4544notbid 317 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑧𝑣𝑧) → (¬ 𝑣𝑅𝑤 ↔ ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4645ralbidva 3166 . . . . . . . . . . . . . . . . . 18 (𝑤𝑧 → (∀𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∀𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4746rexbiia 3082 . . . . . . . . . . . . . . . . 17 (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤))
48 breq1 5148 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧)‘𝑣) = 𝑓 → (((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ 𝑓𝑆((𝐹𝑧)‘𝑤)))
4948notbid 317 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑧)‘𝑣) = 𝑓 → (¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
5049cbvfo 7295 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∀𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
5150rexbidv 3169 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑤𝑧𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
52 breq2 5149 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧)‘𝑤) = 𝑢 → (𝑓𝑆((𝐹𝑧)‘𝑤) ↔ 𝑓𝑆𝑢))
5352notbid 317 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑧)‘𝑤) = 𝑢 → (¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ¬ 𝑓𝑆𝑢))
5453ralbidv 3168 . . . . . . . . . . . . . . . . . . 19 (((𝐹𝑧)‘𝑤) = 𝑢 → (∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5554cbvexfo 7296 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5651, 55bitrd 278 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5747, 56bitrid 282 . . . . . . . . . . . . . . . 16 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5832, 57syl 17 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5931, 58sylibrd 258 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))
6059exp4b 429 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6160com34 91 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ≠ ∅ → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6261com23 86 . . . . . . . . . . 11 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6362imp4a 421 . . . . . . . . . 10 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤)))
6463alrimdv 1925 . . . . . . . . 9 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ∀𝑧((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤)))
65 df-fr 5629 . . . . . . . . 9 (𝑅 Fr dom 𝐹 ↔ ∀𝑧((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))
6664, 65imbitrrdi 251 . . . . . . . 8 (Fun 𝐹 → (𝑆 Fr ran 𝐹𝑅 Fr dom 𝐹))
67 freq2 5645 . . . . . . . . 9 (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹𝑅 Fr 𝐴))
6867biimpd 228 . . . . . . . 8 (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹𝑅 Fr 𝐴))
6966, 68sylan9 506 . . . . . . 7 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝑆 Fr ran 𝐹𝑅 Fr 𝐴))
705, 69sylbi 216 . . . . . 6 (𝐹 Fn 𝐴 → (𝑆 Fr ran 𝐹𝑅 Fr 𝐴))
714, 70sylan9r 507 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
722, 71sylbi 216 . . . 4 (𝐹:𝐴onto𝐵 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
731, 72syl 17 . . 3 (𝐹:𝐴1-1-onto𝐵 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
74 df-f1o 6553 . . . . 5 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
75 fveq2 6893 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
7675breq1d 5155 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑦)))
77 fveq2 6893 . . . . . . . . . . 11 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
7877breq2d 5157 . . . . . . . . . 10 (𝑦 = 𝑣 → ((𝐹𝑤)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑣)))
7934, 33, 76, 78, 39brab 5541 . . . . . . . . 9 (𝑤𝑅𝑣 ↔ (𝐹𝑤)𝑆(𝐹𝑣))
8079a1i 11 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑤𝑅𝑣 ↔ (𝐹𝑤)𝑆(𝐹𝑣)))
81 f1fveq 7269 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → ((𝐹𝑤) = (𝐹𝑣) ↔ 𝑤 = 𝑣))
8281bicomd 222 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑤 = 𝑣 ↔ (𝐹𝑤) = (𝐹𝑣)))
8340a1i 11 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑣𝑅𝑤 ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
8480, 82, 833orbi123d 1432 . . . . . . 7 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → ((𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤))))
85842ralbidva 3207 . . . . . 6 (𝐹:𝐴1-1𝐵 → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤))))
86 breq1 5148 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑤)𝑆(𝐹𝑣) ↔ 𝑢𝑆(𝐹𝑣)))
87 eqeq1 2730 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑤) = (𝐹𝑣) ↔ 𝑢 = (𝐹𝑣)))
88 breq2 5149 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑣)𝑆(𝐹𝑤) ↔ (𝐹𝑣)𝑆𝑢))
8986, 87, 883orbi123d 1432 . . . . . . . . 9 ((𝐹𝑤) = 𝑢 → (((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
9089ralbidv 3168 . . . . . . . 8 ((𝐹𝑤) = 𝑢 → (∀𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
9190cbvfo 7295 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑢𝐵𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
92 breq2 5149 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → (𝑢𝑆(𝐹𝑣) ↔ 𝑢𝑆𝑓))
93 eqeq2 2738 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → (𝑢 = (𝐹𝑣) ↔ 𝑢 = 𝑓))
94 breq1 5148 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → ((𝐹𝑣)𝑆𝑢𝑓𝑆𝑢))
9592, 93, 943orbi123d 1432 . . . . . . . . 9 ((𝐹𝑣) = 𝑓 → ((𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9695cbvfo 7295 . . . . . . . 8 (𝐹:𝐴onto𝐵 → (∀𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ ∀𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9796ralbidv 3168 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∀𝑢𝐵𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9891, 97bitrd 278 . . . . . 6 (𝐹:𝐴onto𝐵 → (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9985, 98sylan9bb 508 . . . . 5 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
10074, 99sylbi 216 . . . 4 (𝐹:𝐴1-1-onto𝐵 → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
101100biimprd 247 . . 3 (𝐹:𝐴1-1-onto𝐵 → (∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢) → ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤)))
10273, 101anim12d 607 . 2 (𝐹:𝐴1-1-onto𝐵 → ((𝑆 Fr 𝐵 ∧ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)) → (𝑅 Fr 𝐴 ∧ ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤))))
103 dfwe2 7774 . 2 (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
104 dfwe2 7774 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤)))
105102, 103, 1043imtr4g 295 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3o 1083  wal 1532   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  wss 3946  c0 4322   class class class wbr 5145  {copab 5207   Fr wfr 5626   We wwe 5628  dom cdm 5674  ran crn 5675  cres 5676  cima 5677  Fun wfun 6540   Fn wfn 6541  1-1wf1 6543  ontowfo 6544  1-1-ontowf1o 6545  cfv 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554
This theorem is referenced by: (None)
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