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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 6855 | . . 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 6855 | . . 3 ⊢ (𝐻:𝐸–1-1-onto→𝐼 → ◡𝐻:𝐼–1-1-onto→𝐸) | |
4 | 3f1oss1 46975 | . . 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 1413 | . 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 6846 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
7 | dfrel2 6205 | . . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
8 | 7 | biimpi 216 | . . . . . . . 8 ⊢ (Rel 𝐹 → ◡◡𝐹 = 𝐹) |
9 | 8 | eqcomd 2739 | . . . . . . 7 ⊢ (Rel 𝐹 → 𝐹 = ◡◡𝐹) |
10 | 9 | coeq2d 5870 | . . . . . 6 ⊢ (Rel 𝐹 → ((◡𝐻 ∘ 𝐺) ∘ 𝐹) = ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹)) |
11 | 10 | f1oeq1d 6838 | . . . . 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 1131 | . . 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 1085 = wceq 1535 ⊆ wss 3963 ◡ccnv 5682 “ cima 5686 ∘ ccom 5687 Rel wrel 5688 –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-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 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-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 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 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 |
This theorem is referenced by: uspgrlim 47817 |
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