Theorem f1ocoima 7299

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 6817	. . . . . 6 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1→𝐷)
2	1	anim1i 626	. . . . 5 ⊢ ((𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
3	2	3adant1 1146	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
4		f1ores 6833	. . . 4 ⊢ ((𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵):𝐵–1-1-onto→(𝐺 “ 𝐵))
5	3, 4	syl 18	. . 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 1152	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1oco 6842	. . 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 595	. 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 6826	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
10		forn 6793	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
11	9, 10	syl 18	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 = 𝐵)
12	11	eqimssd 4001	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 ⊆ 𝐵)
13	12	3ad2ant1 1149	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐵)
14		cores 6248	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ((𝐺 ↾ 𝐵) ∘ 𝐹) = (𝐺 ∘ 𝐹))
15	14	eqcomd 2775	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
16	13, 15	syl 18	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
17	16	f1oeq1d 6813	. 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 260	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 400   ∧ w3a 1101   = wceq 1567   ⊆ wss 3913  ran crn 5660   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  –1-1→wf1 6531  –onto→wfo 6532  –1-1-onto→wf1o 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541

This theorem is referenced by:  3f1oss1  47696