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Theorem fvcofneq 7108
Description: The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)
Assertion
Ref Expression
fvcofneq ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝑥,𝐾   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem fvcofneq
StepHypRef Expression
1 simpl 481 . . . 4 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2 elinel1 4197 . . . . 5 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
323ad2ant1 1130 . . . 4 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐴)
4 fvco2 7000 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
51, 3, 4syl2an 594 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 simpr 483 . . . . 5 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
7 elinel2 4198 . . . . . 6 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
873ad2ant1 1130 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐵)
9 fvco2 7000 . . . . 5 ((𝐾 Fn 𝐵𝑋𝐵) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
106, 8, 9syl2an 594 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
11 fveq2 6902 . . . . . . 7 ((𝐾𝑋) = (𝐺𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1211eqcoms 2736 . . . . . 6 ((𝐺𝑋) = (𝐾𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
13123ad2ant2 1131 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1413adantl 480 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
15 id 22 . . . . . . . . . . . 12 (𝐺 Fn 𝐴𝐺 Fn 𝐴)
16 fnfvelrn 7095 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1715, 2, 16syl2anr 595 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1817ex 411 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐺 Fn 𝐴 → (𝐺𝑋) ∈ ran 𝐺))
19 id 22 . . . . . . . . . . . 12 (𝐾 Fn 𝐵𝐾 Fn 𝐵)
20 fnfvelrn 7095 . . . . . . . . . . . 12 ((𝐾 Fn 𝐵𝑋𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2119, 7, 20syl2anr 595 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2221ex 411 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐾 Fn 𝐵 → (𝐾𝑋) ∈ ran 𝐾))
2318, 22anim12d 607 . . . . . . . . 9 (𝑋 ∈ (𝐴𝐵) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾)))
24 eleq1 2817 . . . . . . . . . . . 12 ((𝐾𝑋) = (𝐺𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2524eqcoms 2736 . . . . . . . . . . 11 ((𝐺𝑋) = (𝐾𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2625anbi2d 628 . . . . . . . . . 10 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾)))
27 elin 3965 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾))
2827biimpri 227 . . . . . . . . . 10 (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
2926, 28biimtrdi 252 . . . . . . . . 9 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
3023, 29sylan9 506 . . . . . . . 8 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
31 fveq2 6902 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
32 fveq2 6902 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐻𝑥) = (𝐻‘(𝐺𝑋)))
3331, 32eqeq12d 2744 . . . . . . . . . . 11 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) = (𝐻𝑥) ↔ (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋))))
3433rspcva 3609 . . . . . . . . . 10 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋)))
3534eqcomd 2734 . . . . . . . . 9 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
3635ex 411 . . . . . . . 8 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋))))
3730, 36syl6 35 . . . . . . 7 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
3837com23 86 . . . . . 6 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
39383impia 1114 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋))))
4039impcom 406 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
4110, 14, 403eqtrrd 2773 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐹‘(𝐺𝑋)) = ((𝐻𝐾)‘𝑋))
425, 41eqtrd 2768 . 2 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4342ex 411 1 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  cin 3948  ran crn 5683  ccom 5686   Fn wfn 6548  cfv 6553
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561
This theorem is referenced by:  fvcosymgeq  19391
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