tposf2 - Intuitionistic Logic Explorer

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Theorem tposf2 6529

Description: The domain and codomain 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 5528	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn4 5616	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
3	1, 2	sylib 122	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
4		tposfo2 6528	. . . . . 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 124	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:^◡𝐴–onto→ran 𝐹)
7		fof 5610	. . . 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 5537	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	9	adantl 277	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵)
11		fss 5541	. . 3 ⊢ ((tpos 𝐹:^◡𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → tpos 𝐹:^◡𝐴⟶𝐵)
12	8, 10, 11	syl2anc 415	. 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:^◡𝐴⟶𝐵)
13	12	ex 115	1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:^◡𝐴⟶𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ⊆ wss 3220 ^◡ccnv 4768 ran crn 4770 Rel wrel 4774 Fn wfn 5367 ⟶wf 5368 –onto→wfo 5370 tpos ctpos 6505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fo 5378 df-fv 5380 df-tpos 6506

This theorem is referenced by: tposf 6533