Theorem f1cofveqaeq 7245

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 487	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹:𝐵–1-1→𝐶)
2		f1f 6764	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		ffvelcdm 7066	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ 𝐵)
4	3	ex 417	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ 𝐵))
5		ffvelcdm 7066	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → (𝐺‘𝑌) ∈ 𝐵)
6	5	ex 417	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑌 ∈ 𝐴 → (𝐺‘𝑌) ∈ 𝐵))
7	4, 6	anim12d 620	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
8	2, 7	syl 18	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
9	8	adantl 486	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵)))
10	9	imp 411	. . 3 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵))
11		f1veqaeq 7244	. . 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 697	. 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 7244	. . 3 ⊢ ((𝐺:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐺‘𝑋) = (𝐺‘𝑌) → 𝑋 = 𝑌))
14	13	adantll 726	. 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 48	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 400   = wceq 1563   ∈ wcel 2145  ⟶wf 6521  –1-1→wf1 6522  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fv 6533

This theorem is referenced by:  uspgrn2crct  30066  upgrimtrlslem2  48525