tposf2 - Intuitionistic Logic Explorer

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Theorem tposf2 6499

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 5508	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn4 5596	. . . . . . 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 6498	. . . . . 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 5590	. . . 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 5517	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	9	adantl 277	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵)
11		fss 5521	. . 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 3211 ^◡ccnv 4748 ran crn 4750 Rel wrel 4754 Fn wfn 5347 ⟶wf 5348 –onto→wfo 5350 tpos ctpos 6475

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fo 5358 df-fv 5360 df-tpos 6476

This theorem is referenced by: tposf 6503