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Mirrors > Home > MPE Home > Th. List > tposfo2 | Structured version Visualization version GIF version |
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposfo2 | ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfn2 8183 | . . . 4 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) | |
2 | 1 | adantrd 493 | . . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn ◡𝐴)) |
3 | fndm 6609 | . . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | releqd 5738 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴)) |
5 | 4 | biimparc 481 | . . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → Rel dom 𝐹) |
6 | rntpos 8174 | . . . . . . 7 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹) |
8 | 7 | eqeq1d 2735 | . . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵)) |
9 | 8 | biimprd 248 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵)) |
10 | 9 | expimpd 455 | . . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵)) |
11 | 2, 10 | jcad 514 | . 2 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵))) |
12 | df-fo 6506 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
13 | df-fo 6506 | . 2 ⊢ (tpos 𝐹:◡𝐴–onto→𝐵 ↔ (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵)) | |
14 | 11, 12, 13 | 3imtr4g 296 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ◡ccnv 5636 dom cdm 5637 ran crn 5638 Rel wrel 5642 Fn wfn 6495 –onto→wfo 6498 tpos ctpos 8160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fo 6506 df-fv 6508 df-tpos 8161 |
This theorem is referenced by: tposf2 8185 tposf1o2 8187 tposfo 8188 oppglsm 19432 |
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