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Theorem f1cofveqaeq 40323
 Description: If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
Assertion
Ref Expression
f1cofveqaeq (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Proof of Theorem f1cofveqaeq
StepHypRef Expression
1 simpl 472 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → 𝐹:𝐵1-1𝐶)
2 f1f 6014 . . . . . 6 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
3 ffvelrn 6265 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → (𝐺𝑋) ∈ 𝐵)
43ex 449 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑋𝐴 → (𝐺𝑋) ∈ 𝐵))
5 ffvelrn 6265 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → (𝐺𝑌) ∈ 𝐵)
65ex 449 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑌𝐴 → (𝐺𝑌) ∈ 𝐵))
74, 6anim12d 584 . . . . . 6 (𝐺:𝐴𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
82, 7syl 17 . . . . 5 (𝐺:𝐴1-1𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
98adantl 481 . . . 4 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
109imp 444 . . 3 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵))
11 f1veqaeq 6418 . . 3 ((𝐹:𝐵1-1𝐶 ∧ ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
121, 10, 11syl2an2r 872 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
13 f1veqaeq 6418 . . 3 ((𝐺:𝐴1-1𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1413adantll 746 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1512, 14syld 46 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  uspgrn2crct  41011
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