Theorem f1cofveqaeq 7131

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 483	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹:𝐵–1-1→𝐶)
2		f1f 6670	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		ffvelrn 6959	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ 𝐵)
4	3	ex 413	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ 𝐵))
5		ffvelrn 6959	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → (𝐺‘𝑌) ∈ 𝐵)
6	5	ex 413	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑌 ∈ 𝐴 → (𝐺‘𝑌) ∈ 𝐵))
7	4, 6	anim12d 609	. . . . . 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 482	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
10	9	imp 407	. . 3 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵))
11		f1veqaeq 7130	. . 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 682	. 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 7130	. . 3 ⊢ ((𝐺:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) = (𝐺‘𝑌) → 𝑋 = 𝑌))
14	13	adantll 711	. 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 396   = wceq 1539   ∈ wcel 2106  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441

This theorem is referenced by:  uspgrn2crct  28173