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Theorem fvcofneq 6840
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 486 . . . 4 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2 elinel1 4155 . . . . 5 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
323ad2ant1 1130 . . . 4 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐴)
4 fvco2 6739 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
51, 3, 4syl2an 598 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 simpr 488 . . . . 5 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
7 elinel2 4156 . . . . . 6 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
873ad2ant1 1130 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐵)
9 fvco2 6739 . . . . 5 ((𝐾 Fn 𝐵𝑋𝐵) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
106, 8, 9syl2an 598 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
11 fveq2 6651 . . . . . . 7 ((𝐾𝑋) = (𝐺𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1211eqcoms 2832 . . . . . 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 485 . . . 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 6829 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1715, 2, 16syl2anr 599 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺𝑋) ∈ ran 𝐺)
1817ex 416 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐺 Fn 𝐴 → (𝐺𝑋) ∈ ran 𝐺))
19 id 22 . . . . . . . . . . . 12 (𝐾 Fn 𝐵𝐾 Fn 𝐵)
20 fnfvelrn 6829 . . . . . . . . . . . 12 ((𝐾 Fn 𝐵𝑋𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2119, 7, 20syl2anr 599 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2221ex 416 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐾 Fn 𝐵 → (𝐾𝑋) ∈ ran 𝐾))
2318, 22anim12d 611 . . . . . . . . 9 (𝑋 ∈ (𝐴𝐵) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾)))
24 eleq1 2903 . . . . . . . . . . . 12 ((𝐾𝑋) = (𝐺𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2524eqcoms 2832 . . . . . . . . . . 11 ((𝐺𝑋) = (𝐾𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2625anbi2d 631 . . . . . . . . . 10 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾)))
27 elin 3934 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾))
2827biimpri 231 . . . . . . . . . 10 (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
2926, 28syl6bi 256 . . . . . . . . 9 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
3023, 29sylan9 511 . . . . . . . 8 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
31 fveq2 6651 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
32 fveq2 6651 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐻𝑥) = (𝐻‘(𝐺𝑋)))
3331, 32eqeq12d 2840 . . . . . . . . . . 11 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) = (𝐻𝑥) ↔ (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋))))
3433rspcva 3606 . . . . . . . . . 10 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋)))
3534eqcomd 2830 . . . . . . . . 9 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
3635ex 416 . . . . . . . 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 411 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
4110, 14, 403eqtrrd 2864 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐹‘(𝐺𝑋)) = ((𝐻𝐾)‘𝑋))
425, 41eqtrd 2859 . 2 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4342ex 416 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  cin 3917  ran crn 5537  ccom 5540   Fn wfn 6331  cfv 6336
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344
This theorem is referenced by:  fvcosymgeq  18546
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