tposf2 - Intuitionistic Logic Explorer

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Theorem tposf2 6335

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 5410	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn4 5489	. . . . . . 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 6334	. . . . . 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 5483	. . . 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 5419	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	9	adantl 277	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵)
11		fss 5422	. . 3 ⊢ ((tpos 𝐹:^◡𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → tpos 𝐹:^◡𝐴⟶𝐵)
12	8, 10, 11	syl2anc 411	. 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 3157 ^◡ccnv 4663 ran crn 4665 Rel wrel 4669 Fn wfn 5254 ⟶wf 5255 –onto→wfo 5257 tpos ctpos 6311

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fo 5265 df-fv 5267 df-tpos 6312

This theorem is referenced by: tposf 6339