tposfn2 - Metamath Proof Explorer

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Theorem tposfn2 8245

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 8239	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 8235	. . . . . 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 5772	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5870	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ^◡dom 𝐹 = ^◡𝐴)
7	6	eqeq2d 2738	. . . . 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 608	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴)))
11		df-fn 6545	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 6545	. 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 395 = wceq 1534 ^◡ccnv 5671 dom cdm 5672 Rel wrel 5677 Fun wfun 6536 Fn wfn 6537 tpos ctpos 8222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-tpos 8223

This theorem is referenced by: tposfo2 8246 tpos0 8253