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Theorem f1cofveqaeq 7260
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 482 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → 𝐹:𝐵1-1𝐶)
2 f1f 6787 . . . . . 6 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
3 ffvelcdm 7083 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → (𝐺𝑋) ∈ 𝐵)
43ex 412 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑋𝐴 → (𝐺𝑋) ∈ 𝐵))
5 ffvelcdm 7083 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → (𝐺𝑌) ∈ 𝐵)
65ex 412 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑌𝐴 → (𝐺𝑌) ∈ 𝐵))
74, 6anim12d 608 . . . . . 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 406 . . 3 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵))
11 f1veqaeq 7259 . . 3 ((𝐹:𝐵1-1𝐶 ∧ ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
121, 10, 11syl2an2r 682 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
13 f1veqaeq 7259 . . 3 ((𝐺:𝐴1-1𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1413adantll 711 . 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 395   = wceq 1540  wcel 2105  wf 6539  1-1wf1 6540  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  uspgrn2crct  29495
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