Theorem f1ocoima 7342

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⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵))

Proof of Theorem f1ocoima

Step	Hyp	Ref	Expression
1		f1of1 6853	. . . . . 6 ⊢ (𝐺:𝐶⟶1-1onto𝐷 → 𝐺:𝐶⟶1-1𝐷)
2	1	anim1i 627	. . . . 5 ⊢ ((𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶⟶1-1𝐷 ∧ 𝐵 ⊆ 𝐶))
3	2	3adant1 1148	. . . 4 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶⟶1-1𝐷 ∧ 𝐵 ⊆ 𝐶))
4		f1ores 6869	. . . 4 ⊢ ((𝐺:𝐶⟶1-1𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵):𝐵⟶1-1onto(𝐺 “ 𝐵))
5	3, 4	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵):𝐵⟶1-1onto(𝐺 “ 𝐵))
6		simp1 1154	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶1-1onto𝐵)
7		f1oco 6878	. . 3 ⊢ (((𝐺 ↾ 𝐵):𝐵⟶1-1onto(𝐺 “ 𝐵) ∧ 𝐹:𝐴⟶1-1onto𝐵) → ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵))
8	5, 6, 7	syl2anc 596	. 2 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵))
9		f1ofo 6862	. . . . . . 7 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶onto𝐵)
10		forn 6829	. . . . . . 7 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
11	9, 10	syl 18	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → ran 𝐹 = 𝐵)
12	11	eqimssd 3969	. . . . 5 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → ran 𝐹 ⊆ 𝐵)
13	12	3ad2ant1 1151	. . . 4 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐵)
14		cores 6269	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ((𝐺 ↾ 𝐵) ∘ 𝐹) = (𝐺 ∘ 𝐹))
15	14	eqcomd 2769	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
16	13, 15	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹) = ((𝐺 ↾ 𝐵) ∘ 𝐹))
17	16	f1oeq1d 6849	. 2 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → ((𝐺 ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵) ↔ ((𝐺 ↾ 𝐵) ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵)))
18	8, 17	mpbird 260	1 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴⟶1-1onto(𝐺 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ⊆ wss 3900  ran crn 5659   ↾ cres 5660   “ cima 5661   ∘ ccom 5662  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  3f1oss1  48418