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Theorem fvcofneq 6631
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 476 . . . 4 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2 elinel1 4022 . . . . 5 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
323ad2ant1 1124 . . . 4 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐴)
4 fvco2 6533 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
51, 3, 4syl2an 589 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 simpr 479 . . . . 5 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
7 elinel2 4023 . . . . . 6 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
873ad2ant1 1124 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐵)
9 fvco2 6533 . . . . 5 ((𝐾 Fn 𝐵𝑋𝐵) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
106, 8, 9syl2an 589 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
11 fveq2 6446 . . . . . . 7 ((𝐾𝑋) = (𝐺𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1211eqcoms 2786 . . . . . 6 ((𝐺𝑋) = (𝐾𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
13123ad2ant2 1125 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1413adantl 475 . . . 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 6620 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1715, 2, 16syl2anr 590 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1817ex 403 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐺 Fn 𝐴 → (𝐺𝑋) ∈ ran 𝐺))
19 id 22 . . . . . . . . . . . 12 (𝐾 Fn 𝐵𝐾 Fn 𝐵)
20 fnfvelrn 6620 . . . . . . . . . . . 12 ((𝐾 Fn 𝐵𝑋𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2119, 7, 20syl2anr 590 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2221ex 403 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐾 Fn 𝐵 → (𝐾𝑋) ∈ ran 𝐾))
2318, 22anim12d 602 . . . . . . . . 9 (𝑋 ∈ (𝐴𝐵) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾)))
24 eleq1 2847 . . . . . . . . . . . 12 ((𝐾𝑋) = (𝐺𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2524eqcoms 2786 . . . . . . . . . . 11 ((𝐺𝑋) = (𝐾𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2625anbi2d 622 . . . . . . . . . 10 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾)))
27 elin 4019 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾))
2827biimpri 220 . . . . . . . . . 10 (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
2926, 28syl6bi 245 . . . . . . . . 9 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
3023, 29sylan9 503 . . . . . . . 8 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
31 fveq2 6446 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
32 fveq2 6446 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐻𝑥) = (𝐻‘(𝐺𝑋)))
3331, 32eqeq12d 2793 . . . . . . . . . . 11 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) = (𝐻𝑥) ↔ (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋))))
3433rspcva 3509 . . . . . . . . . 10 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋)))
3534eqcomd 2784 . . . . . . . . 9 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
3635ex 403 . . . . . . . 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 1106 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋))))
4039impcom 398 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
4110, 14, 403eqtrrd 2819 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐹‘(𝐺𝑋)) = ((𝐻𝐾)‘𝑋))
425, 41eqtrd 2814 . 2 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4342ex 403 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  cin 3791  ran crn 5356  ccom 5359   Fn wfn 6130  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143
This theorem is referenced by:  fvcosymgeq  18232
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