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Mirrors > Home > MPE Home > Th. List > f1ocoima | Structured version Visualization version GIF version |
Description: The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.) |
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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of1 6842 | . . . . . 6 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1→𝐷) | |
2 | 1 | anim1i 614 | . . . . 5 ⊢ ((𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶)) |
3 | 2 | 3adant1 1128 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐵 ⊆ 𝐶)) |
4 | f1ores 6857 | . . . 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 1134 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1-onto→𝐵) | |
7 | f1oco 6866 | . . 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 583 | . 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 6850 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
10 | forn 6818 | . . . . . . 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 4052 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 ⊆ 𝐵) |
13 | 12 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐵) |
14 | cores 6265 | . . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ((𝐺 ↾ 𝐵) ∘ 𝐹) = (𝐺 ∘ 𝐹)) | |
15 | 14 | eqcomd 2739 | . . . 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 6838 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ⊆ wss 3963 ran crn 5684 ↾ cres 5685 “ cima 5686 ∘ ccom 5687 –1-1→wf1 6555 –onto→wfo 6556 –1-1-onto→wf1o 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 |
This theorem is referenced by: 3f1oss1 46975 |
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