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Theorem tfrlem5 7999
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem5 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,,𝑢,𝑣,𝐹   𝐴,𝑔,
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓)

Proof of Theorem tfrlem5
Dummy variables 𝑧 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 vex 3444 . . 3 𝑔 ∈ V
31, 2tfrlem3a 7996 . 2 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
4 vex 3444 . . 3 ∈ V
51, 4tfrlem3a 7996 . 2 (𝐴 ↔ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎))))
6 reeanv 3320 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ↔ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))))
7 fveq2 6645 . . . . . . . 8 (𝑎 = 𝑥 → (𝑔𝑎) = (𝑔𝑥))
8 fveq2 6645 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎) = (𝑥))
97, 8eqeq12d 2814 . . . . . . 7 (𝑎 = 𝑥 → ((𝑔𝑎) = (𝑎) ↔ (𝑔𝑥) = (𝑥)))
10 onin 6190 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤) ∈ On)
11103ad2ant1 1130 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ∈ On)
12 simp2ll 1237 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑔 Fn 𝑧)
13 fnfun 6423 . . . . . . . . . 10 (𝑔 Fn 𝑧 → Fun 𝑔)
1412, 13syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun 𝑔)
15 inss1 4155 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑧
1612fndmd 6427 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom 𝑔 = 𝑧)
1715, 16sseqtrrid 3968 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom 𝑔)
1814, 17jca 515 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun 𝑔 ∧ (𝑧𝑤) ⊆ dom 𝑔))
19 simp2rl 1239 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fn 𝑤)
20 fnfun 6423 . . . . . . . . . 10 ( Fn 𝑤 → Fun )
2119, 20syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun )
22 inss2 4156 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑤
2319fndmd 6427 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom = 𝑤)
2422, 23sseqtrrid 3968 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom )
2521, 24jca 515 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun ∧ (𝑧𝑤) ⊆ dom ))
26 simp2lr 1238 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)))
27 ssralv 3981 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑧 → (∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎))))
2815, 26, 27mpsyl 68 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎)))
29 simp2rr 1240 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))
30 ssralv 3981 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑤 → (∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎))))
3122, 29, 30mpsyl 68 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎)))
3211, 18, 25, 28, 31tfrlem1 7995 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎))
33 simp3l 1198 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑔𝑢)
34 fnbr 6430 . . . . . . . . 9 ((𝑔 Fn 𝑧𝑥𝑔𝑢) → 𝑥𝑧)
3512, 33, 34syl2anc 587 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑧)
36 simp3r 1199 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑣)
37 fnbr 6430 . . . . . . . . 9 (( Fn 𝑤𝑥𝑣) → 𝑥𝑤)
3819, 36, 37syl2anc 587 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑤)
3935, 38elind 4121 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥 ∈ (𝑧𝑤))
409, 32, 39rspcdva 3573 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = (𝑥))
41 funbrfv 6691 . . . . . . 7 (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔𝑥) = 𝑢))
4214, 33, 41sylc 65 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = 𝑢)
43 funbrfv 6691 . . . . . . 7 (Fun → (𝑥𝑣 → (𝑥) = 𝑣))
4421, 36, 43sylc 65 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥) = 𝑣)
4540, 42, 443eqtr3d 2841 . . . . 5 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑢 = 𝑣)
46453exp 1116 . . . 4 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)))
4746rexlimivv 3251 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
486, 47sylbir 238 . 2 ((∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
493, 5, 48syl2anb 600 1 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  {cab 2776  wral 3106  wrex 3107  cin 3880  wss 3881   class class class wbr 5030  dom cdm 5519  cres 5521  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  cfv 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332
This theorem is referenced by:  tfrlem7  8002
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