tposfn2 - Metamath Proof Explorer

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Theorem tposfn2 8064

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

**Assertion**
Ref	Expression
tposfn2	⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

**Proof of Theorem tposfn2**
Step	Hyp	Ref	Expression
1		tposfun 8058	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 8054	. . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ^◡dom 𝐹)
4	3	a1i 11	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = ^◡dom 𝐹))
5		releq 5687	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5782	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ^◡dom 𝐹 = ^◡𝐴)
7	6	eqeq2d 2749	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ^◡dom 𝐹 ↔ dom tpos 𝐹 = ^◡𝐴))
8	4, 5, 7	3imtr3d 293	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = ^◡𝐴))
9	8	com12 32	. . 3 ⊢ (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = ^◡𝐴))
10	2, 9	anim12d 609	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴)))
11		df-fn 6436	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 6436	. 2 ⊢ (tpos 𝐹 Fn ^◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴))
13	10, 11, 12	3imtr4g 296	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ^◡ccnv 5588 dom cdm 5589 Rel wrel 5594 Fun wfun 6427 Fn wfn 6428 tpos ctpos 8041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-fv 6441 df-tpos 8042

This theorem is referenced by: tposfo2 8065 tpos0 8072