Theorem f1ocoima 7251

Description: The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.)

Assertion

Ref	Expression
f1ocoima	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))

Proof of Theorem f1ocoima

Step	Hyp	Ref	Expression
1		f1of1 6770	. . . . . 6 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1→𝐷)
2	1	anim1i 622	. . . . 5 ⊢ ((𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
3	2	3adant1 1137	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
4		f1ores 6785	. . . 4 ⊢ ((𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵):𝐵–1-1-onto→(𝐺 “ 𝐵))
5	3, 4	syl 17	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵):𝐵–1-1-onto→(𝐺 “ 𝐵))
6		simp1 1143	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1oco 6794	. . 3 ⊢ (((𝐺 ↾ 𝐵):𝐵–1-1-onto→(𝐺 “ 𝐵) ∧ 𝐹:𝐴–1-1-onto→𝐵) → ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))
8	5, 6, 7	syl2anc 591	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))
9		f1ofo 6778	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
10		forn 6746	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 = 𝐵)
12	11	eqimssd 3973	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 ⊆ 𝐵)
13	12	3ad2ant1 1140	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐵)
14		cores 6204	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ((𝐺 ↾ 𝐵) ∘ 𝐹) = (𝐺 ∘ 𝐹))
15	14	eqcomd 2747	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
16	13, 15	syl 17	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
17	16	f1oeq1d 6766	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵) ↔ ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵)))
18	8, 17	mpbird 259	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ⊆ wss 3885  ran crn 5622   ↾ cres 5623   “ cima 5624   ∘ ccom 5625  –1-1→wf1 6486  –onto→wfo 6487  –1-1-onto→wf1o 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by:  3f1oss1  47552