Theorem f1ocoima 7280

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 6801	. . . . . 6 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1→𝐷)
2	1	anim1i 615	. . . . 5 ⊢ ((𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
3	2	3adant1 1130	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶))
4		f1ores 6816	. . . 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 1136	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1oco 6825	. . 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 584	. 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 6809	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
10		forn 6777	. . . . . . 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 4005	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 ⊆ 𝐵)
13	12	3ad2ant1 1133	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐵)
14		cores 6224	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ((𝐺 ↾ 𝐵) ∘ 𝐹) = (𝐺 ∘ 𝐹))
15	14	eqcomd 2736	. . . 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 6797	. 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 257	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 395   ∧ w3a 1086   = wceq 1540   ⊆ wss 3916  ran crn 5641   ↾ cres 5642   “ cima 5643   ∘ ccom 5644  –1-1→wf1 6510  –onto→wfo 6511  –1-1-onto→wf1o 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520

This theorem is referenced by:  3f1oss1  47066