tposf2 - Intuitionistic Logic Explorer

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Theorem tposf2 5906

Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)

**Assertion**
Ref	Expression
tposf2	⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:^◡𝐴⟶𝐵))

**Proof of Theorem tposf2**
Step	Hyp	Ref	Expression
1		ffn 5066	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn4 5132	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
3	1, 2	sylib 120	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
4		tposfo2 5905	. . . . . 6 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→ran 𝐹 → tpos 𝐹:^◡𝐴–onto→ran 𝐹))
5	3, 4	syl5 32	. . . . 5 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:^◡𝐴–onto→ran 𝐹))
6	5	imp 122	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:^◡𝐴–onto→ran 𝐹)
7		fof 5126	. . . 4 ⊢ (tpos 𝐹:^◡𝐴–onto→ran 𝐹 → tpos 𝐹:^◡𝐴⟶ran 𝐹)
8	6, 7	syl 14	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:^◡𝐴⟶ran 𝐹)
9		frn 5072	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	9	adantl 271	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵)
11		fss 5074	. . 3 ⊢ ((tpos 𝐹:^◡𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → tpos 𝐹:^◡𝐴⟶𝐵)
12	8, 10, 11	syl2anc 403	. 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:^◡𝐴⟶𝐵)
13	12	ex 113	1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:^◡𝐴⟶𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ⊆ wss 2973 ^◡ccnv 4362 ran crn 4364 Rel wrel 4368 Fn wfn 4917 ⟶wf 4918 –onto→wfo 4920 tpos ctpos 5882

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-tpos 5883

This theorem is referenced by: tposf 5910