Theorem f1cofveqaeq 7255

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 6774	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		ffvelcdm 7076	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ 𝐵)
4	3	ex 417	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐵 → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ 𝐵))
5		ffvelcdm 7076	. . . . . . . 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 7254	. . 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 7254	. . 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is used by:  uspgrn2crct  30168  upgrimtrlslem2  48698