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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3f1oss2 | Structured version Visualization version GIF version |
Description: The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.) |
Ref | Expression |
---|---|
3f1oss2 | ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 6858 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | id 22 | . . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1-onto→𝐷) | |
3 | f1ocnv 6858 | . . 3 ⊢ (𝐻:𝐸–1-1-onto→𝐼 → ◡𝐻:𝐼–1-1-onto→𝐸) | |
4 | 3f1oss1 47060 | . . 3 ⊢ (((◡𝐹:𝐵–1-1-onto→𝐴 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ ◡𝐻:𝐼–1-1-onto→𝐸) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)) | |
5 | 1, 2, 3, 4 | syl3anl 1417 | . 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)) |
6 | f1orel 6849 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
7 | dfrel2 6207 | . . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
8 | 7 | biimpi 216 | . . . . . . . 8 ⊢ (Rel 𝐹 → ◡◡𝐹 = 𝐹) |
9 | 8 | eqcomd 2742 | . . . . . . 7 ⊢ (Rel 𝐹 → 𝐹 = ◡◡𝐹) |
10 | 9 | coeq2d 5871 | . . . . . 6 ⊢ (Rel 𝐹 → ((◡𝐻 ∘ 𝐺) ∘ 𝐹) = ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹)) |
11 | 10 | f1oeq1d 6841 | . . . . 5 ⊢ (Rel 𝐹 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))) |
12 | 6, 11 | syl 17 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))) |
13 | 12 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))) |
14 | 13 | adantr 480 | . 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))) |
15 | 5, 14 | mpbird 257 | 1 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ⊆ wss 3950 ◡ccnv 5682 “ cima 5686 ∘ ccom 5687 Rel wrel 5688 –1-1-onto→wf1o 6558 |
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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 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-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 |
This theorem is referenced by: uspgrlim 47932 |
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