tposf2 - Intuitionistic Logic Explorer

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Theorem tposf2 6323

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 5404	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn4 5483	. . . . . . 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 6322	. . . . . 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 5477	. . . 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 5413	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	9	adantl 277	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵)
11		fss 5416	. . 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 3154 ^◡ccnv 4659 ran crn 4661 Rel wrel 4665 Fn wfn 5250 ⟶wf 5251 –onto→wfo 5253 tpos ctpos 6299

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fo 5261 df-fv 5263 df-tpos 6300

This theorem is referenced by: tposf 6327