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Theorem f1cofveqaeq 7206
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 484 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → 𝐹:𝐵1-1𝐶)
2 f1f 6739 . . . . . 6 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
3 ffvelcdm 7033 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → (𝐺𝑋) ∈ 𝐵)
43ex 414 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑋𝐴 → (𝐺𝑋) ∈ 𝐵))
5 ffvelcdm 7033 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → (𝐺𝑌) ∈ 𝐵)
65ex 414 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑌𝐴 → (𝐺𝑌) ∈ 𝐵))
74, 6anim12d 610 . . . . . 6 (𝐺:𝐴𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
82, 7syl 17 . . . . 5 (𝐺:𝐴1-1𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
98adantl 483 . . . 4 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
109imp 408 . . 3 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵))
11 f1veqaeq 7205 . . 3 ((𝐹:𝐵1-1𝐶 ∧ ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
121, 10, 11syl2an2r 684 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
13 f1veqaeq 7205 . . 3 ((𝐺:𝐴1-1𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1413adantll 713 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1512, 14syld 47 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wf 6493  1-1wf1 6494  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505
This theorem is referenced by:  uspgrn2crct  28795
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