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Theorem wemaplem2 9003
 Description: Lemma for wemapso 9007. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Hypotheses
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
wemaplem2.a (𝜑𝐴 ∈ V)
wemaplem2.p (𝜑𝑃 ∈ (𝐵m 𝐴))
wemaplem2.x (𝜑𝑋 ∈ (𝐵m 𝐴))
wemaplem2.q (𝜑𝑄 ∈ (𝐵m 𝐴))
wemaplem2.r (𝜑𝑅 Or 𝐴)
wemaplem2.s (𝜑𝑆 Po 𝐵)
wemaplem2.px1 (𝜑𝑎𝐴)
wemaplem2.px2 (𝜑 → (𝑃𝑎)𝑆(𝑋𝑎))
wemaplem2.px3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
wemaplem2.xq1 (𝜑𝑏𝐴)
wemaplem2.xq2 (𝜑 → (𝑋𝑏)𝑆(𝑄𝑏))
wemaplem2.xq3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
Assertion
Ref Expression
wemaplem2 (𝜑𝑃𝑇𝑄)
Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝐵   𝑇,𝑎,𝑏,𝑐   𝑤,𝑎,𝑦,𝑧,𝑋,𝑏,𝑐,𝑥   𝐴,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑐)   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemaplem2
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 wemaplem2.px1 . . . 4 (𝜑𝑎𝐴)
2 wemaplem2.xq1 . . . 4 (𝜑𝑏𝐴)
31, 2ifcld 4514 . . 3 (𝜑 → if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴)
4 wemaplem2.px2 . . . . . . 7 (𝜑 → (𝑃𝑎)𝑆(𝑋𝑎))
54adantr 481 . . . . . 6 ((𝜑𝑎𝑅𝑏) → (𝑃𝑎)𝑆(𝑋𝑎))
6 breq1 5065 . . . . . . . . 9 (𝑐 = 𝑎 → (𝑐𝑅𝑏𝑎𝑅𝑏))
7 fveq2 6666 . . . . . . . . . 10 (𝑐 = 𝑎 → (𝑋𝑐) = (𝑋𝑎))
8 fveq2 6666 . . . . . . . . . 10 (𝑐 = 𝑎 → (𝑄𝑐) = (𝑄𝑎))
97, 8eqeq12d 2841 . . . . . . . . 9 (𝑐 = 𝑎 → ((𝑋𝑐) = (𝑄𝑐) ↔ (𝑋𝑎) = (𝑄𝑎)))
106, 9imbi12d 346 . . . . . . . 8 (𝑐 = 𝑎 → ((𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)) ↔ (𝑎𝑅𝑏 → (𝑋𝑎) = (𝑄𝑎))))
11 wemaplem2.xq3 . . . . . . . 8 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
1210, 11, 1rspcdva 3628 . . . . . . 7 (𝜑 → (𝑎𝑅𝑏 → (𝑋𝑎) = (𝑄𝑎)))
1312imp 407 . . . . . 6 ((𝜑𝑎𝑅𝑏) → (𝑋𝑎) = (𝑄𝑎))
145, 13breqtrd 5088 . . . . 5 ((𝜑𝑎𝑅𝑏) → (𝑃𝑎)𝑆(𝑄𝑎))
15 iftrue 4475 . . . . . . . 8 (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
1615fveq2d 6670 . . . . . . 7 (𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑎))
1715fveq2d 6670 . . . . . . 7 (𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑎))
1816, 17breq12d 5075 . . . . . 6 (𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
1918adantl 482 . . . . 5 ((𝜑𝑎𝑅𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
2014, 19mpbird 258 . . . 4 ((𝜑𝑎𝑅𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
21 wemaplem2.s . . . . . . 7 (𝜑𝑆 Po 𝐵)
2221adantr 481 . . . . . 6 ((𝜑𝑎 = 𝑏) → 𝑆 Po 𝐵)
23 wemaplem2.p . . . . . . . . . 10 (𝜑𝑃 ∈ (𝐵m 𝐴))
24 elmapi 8421 . . . . . . . . . 10 (𝑃 ∈ (𝐵m 𝐴) → 𝑃:𝐴𝐵)
2523, 24syl 17 . . . . . . . . 9 (𝜑𝑃:𝐴𝐵)
2625, 2ffvelrnd 6847 . . . . . . . 8 (𝜑 → (𝑃𝑏) ∈ 𝐵)
27 wemaplem2.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵m 𝐴))
28 elmapi 8421 . . . . . . . . . 10 (𝑋 ∈ (𝐵m 𝐴) → 𝑋:𝐴𝐵)
2927, 28syl 17 . . . . . . . . 9 (𝜑𝑋:𝐴𝐵)
3029, 2ffvelrnd 6847 . . . . . . . 8 (𝜑 → (𝑋𝑏) ∈ 𝐵)
31 wemaplem2.q . . . . . . . . . 10 (𝜑𝑄 ∈ (𝐵m 𝐴))
32 elmapi 8421 . . . . . . . . . 10 (𝑄 ∈ (𝐵m 𝐴) → 𝑄:𝐴𝐵)
3331, 32syl 17 . . . . . . . . 9 (𝜑𝑄:𝐴𝐵)
3433, 2ffvelrnd 6847 . . . . . . . 8 (𝜑 → (𝑄𝑏) ∈ 𝐵)
3526, 30, 343jca 1122 . . . . . . 7 (𝜑 → ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵))
3635adantr 481 . . . . . 6 ((𝜑𝑎 = 𝑏) → ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵))
37 fveq2 6666 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑃𝑎) = (𝑃𝑏))
38 fveq2 6666 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑋𝑎) = (𝑋𝑏))
3937, 38breq12d 5075 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑃𝑎)𝑆(𝑋𝑎) ↔ (𝑃𝑏)𝑆(𝑋𝑏)))
404, 39syl5ibcom 246 . . . . . . 7 (𝜑 → (𝑎 = 𝑏 → (𝑃𝑏)𝑆(𝑋𝑏)))
4140imp 407 . . . . . 6 ((𝜑𝑎 = 𝑏) → (𝑃𝑏)𝑆(𝑋𝑏))
42 wemaplem2.xq2 . . . . . . 7 (𝜑 → (𝑋𝑏)𝑆(𝑄𝑏))
4342adantr 481 . . . . . 6 ((𝜑𝑎 = 𝑏) → (𝑋𝑏)𝑆(𝑄𝑏))
44 potr 5484 . . . . . . 7 ((𝑆 Po 𝐵 ∧ ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵)) → (((𝑃𝑏)𝑆(𝑋𝑏) ∧ (𝑋𝑏)𝑆(𝑄𝑏)) → (𝑃𝑏)𝑆(𝑄𝑏)))
4544imp 407 . . . . . 6 (((𝑆 Po 𝐵 ∧ ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵)) ∧ ((𝑃𝑏)𝑆(𝑋𝑏) ∧ (𝑋𝑏)𝑆(𝑄𝑏))) → (𝑃𝑏)𝑆(𝑄𝑏))
4622, 36, 41, 43, 45syl22anc 836 . . . . 5 ((𝜑𝑎 = 𝑏) → (𝑃𝑏)𝑆(𝑄𝑏))
47 ifeq1 4473 . . . . . . . . 9 (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = if(𝑎𝑅𝑏, 𝑏, 𝑏))
48 ifid 4508 . . . . . . . . 9 if(𝑎𝑅𝑏, 𝑏, 𝑏) = 𝑏
4947, 48syl6eq 2876 . . . . . . . 8 (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
5049fveq2d 6670 . . . . . . 7 (𝑎 = 𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑏))
5149fveq2d 6670 . . . . . . 7 (𝑎 = 𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑏))
5250, 51breq12d 5075 . . . . . 6 (𝑎 = 𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
5352adantl 482 . . . . 5 ((𝜑𝑎 = 𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
5446, 53mpbird 258 . . . 4 ((𝜑𝑎 = 𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
55 breq1 5065 . . . . . . . . 9 (𝑐 = 𝑏 → (𝑐𝑅𝑎𝑏𝑅𝑎))
56 fveq2 6666 . . . . . . . . . 10 (𝑐 = 𝑏 → (𝑃𝑐) = (𝑃𝑏))
57 fveq2 6666 . . . . . . . . . 10 (𝑐 = 𝑏 → (𝑋𝑐) = (𝑋𝑏))
5856, 57eqeq12d 2841 . . . . . . . . 9 (𝑐 = 𝑏 → ((𝑃𝑐) = (𝑋𝑐) ↔ (𝑃𝑏) = (𝑋𝑏)))
5955, 58imbi12d 346 . . . . . . . 8 (𝑐 = 𝑏 → ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ↔ (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑋𝑏))))
60 wemaplem2.px3 . . . . . . . 8 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
6159, 60, 2rspcdva 3628 . . . . . . 7 (𝜑 → (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑋𝑏)))
6261imp 407 . . . . . 6 ((𝜑𝑏𝑅𝑎) → (𝑃𝑏) = (𝑋𝑏))
6342adantr 481 . . . . . 6 ((𝜑𝑏𝑅𝑎) → (𝑋𝑏)𝑆(𝑄𝑏))
6462, 63eqbrtrd 5084 . . . . 5 ((𝜑𝑏𝑅𝑎) → (𝑃𝑏)𝑆(𝑄𝑏))
65 wemaplem2.r . . . . . . . . 9 (𝜑𝑅 Or 𝐴)
66 sopo 5490 . . . . . . . . 9 (𝑅 Or 𝐴𝑅 Po 𝐴)
6765, 66syl 17 . . . . . . . 8 (𝜑𝑅 Po 𝐴)
68 po2nr 5485 . . . . . . . 8 ((𝑅 Po 𝐴 ∧ (𝑏𝐴𝑎𝐴)) → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏))
6967, 2, 1, 68syl12anc 834 . . . . . . 7 (𝜑 → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏))
70 nan 827 . . . . . . 7 ((𝜑 → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏)) ↔ ((𝜑𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏))
7169, 70mpbi 231 . . . . . 6 ((𝜑𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏)
72 iffalse 4478 . . . . . . . 8 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
7372fveq2d 6670 . . . . . . 7 𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑏))
7472fveq2d 6670 . . . . . . 7 𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑏))
7573, 74breq12d 5075 . . . . . 6 𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
7671, 75syl 17 . . . . 5 ((𝜑𝑏𝑅𝑎) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
7764, 76mpbird 258 . . . 4 ((𝜑𝑏𝑅𝑎) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
78 solin 5496 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
7965, 1, 2, 78syl12anc 834 . . . 4 (𝜑 → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
8020, 54, 77, 79mpjao3dan 1425 . . 3 (𝜑 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
81 r19.26 3174 . . . . 5 (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) ↔ (∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
8260, 11, 81sylanbrc 583 . . . 4 (𝜑 → ∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
8365, 1, 23jca 1122 . . . . 5 (𝜑 → (𝑅 Or 𝐴𝑎𝐴𝑏𝐴))
84 anim12 805 . . . . . . 7 (((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ((𝑐𝑅𝑎𝑐𝑅𝑏) → ((𝑃𝑐) = (𝑋𝑐) ∧ (𝑋𝑐) = (𝑄𝑐))))
85 eqtr 2845 . . . . . . 7 (((𝑃𝑐) = (𝑋𝑐) ∧ (𝑋𝑐) = (𝑄𝑐)) → (𝑃𝑐) = (𝑄𝑐))
8684, 85syl6 35 . . . . . 6 (((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)))
8786ralimi 3164 . . . . 5 (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ∀𝑐𝐴 ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)))
88 simpl1 1185 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑅 Or 𝐴)
89 simpr 485 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑐𝐴)
90 simpl2 1186 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑎𝐴)
91 simpl3 1187 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑏𝐴)
92 soltmin 5993 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑐𝐴𝑎𝐴𝑏𝐴)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎𝑐𝑅𝑏)))
9388, 89, 90, 91, 92syl13anc 1366 . . . . . . . 8 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎𝑐𝑅𝑏)))
9493biimpd 230 . . . . . . 7 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑎𝑐𝑅𝑏)))
9594imim1d 82 . . . . . 6 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
9695ralimdva 3181 . . . . 5 ((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) → (∀𝑐𝐴 ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)) → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
9783, 87, 96syl2im 40 . . . 4 (𝜑 → (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
9882, 97mpd 15 . . 3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))
99 fveq2 6666 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑑) = (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
100 fveq2 6666 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑄𝑑) = (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
10199, 100breq12d 5075 . . . . 5 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑃𝑑)𝑆(𝑄𝑑) ↔ (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))))
102 breq2 5066 . . . . . . 7 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑑𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏)))
103102imbi1d 343 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)) ↔ (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
104103ralbidv 3201 . . . . 5 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)) ↔ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
105101, 104anbi12d 630 . . . 4 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))) ↔ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))))
106105rspcev 3626 . . 3 ((if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴 ∧ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))) → ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))))
1073, 80, 98, 106syl12anc 834 . 2 (𝜑 → ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))))
108 wemapso.t . . . 4 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
109108wemaplem1 9002 . . 3 ((𝑃 ∈ (𝐵m 𝐴) ∧ 𝑄 ∈ (𝐵m 𝐴)) → (𝑃𝑇𝑄 ↔ ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)))))
11023, 31, 109syl2anc 584 . 2 (𝜑 → (𝑃𝑇𝑄 ↔ ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)))))
111107, 110mpbird 258 1 (𝜑𝑃𝑇𝑄)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1080   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∀wral 3142  ∃wrex 3143  Vcvv 3499  ifcif 4469   class class class wbr 5062  {copab 5124   Po wpo 5470   Or wor 5471  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151   ↑m cmap 8399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-po 5472  df-so 5473  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401 This theorem is referenced by:  wemaplem3  9004
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