Theorem f1cofveqaeq 7262

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:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌))

Proof of Theorem f1cofveqaeq

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹:𝐵–1-1→𝐶)
2		f1f 6787	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		ffvelcdm 7085	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ 𝐵)
4	3	ex 412	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ 𝐵))
5		ffvelcdm 7085	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → (𝐺‘𝑌) ∈ 𝐵)
6	5	ex 412	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑌 ∈ 𝐴 → (𝐺‘𝑌) ∈ 𝐵))
7	4, 6	anim12d 608	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
8	2, 7	syl 17	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
9	8	adantl 481	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
10	9	imp 406	. . 3 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵))
11		f1veqaeq 7261	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → (𝐺‘𝑋) = (𝐺‘𝑌)))
12	1, 10, 11	syl2an2r 684	. 2 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → (𝐺‘𝑋) = (𝐺‘𝑌)))
13		f1veqaeq 7261	. . 3 ⊢ ((𝐺:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) = (𝐺‘𝑌) → 𝑋 = 𝑌))
14	13	adantll 713	. 2 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) = (𝐺‘𝑌) → 𝑋 = 𝑌))
15	12, 14	syld 47	1 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ⟶wf 6538  –1-1→wf1 6539  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fv 6550

This theorem is referenced by:  uspgrn2crct  29593