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Theorem cantnfs 9735

Description: Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)

**Hypotheses**
Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)

**Assertion**
Ref	Expression
cantnfs	⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))

Proof of Theorem cantnfs Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		eqid 2740	. . . . . 6 ⊢ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}
3		cantnfs.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
4		cantnfs.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On)
5	2, 3, 4	cantnfdm 9733	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	eqtrid 2792	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2830	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 5169	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3708	. . 3 ⊢ (𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅} ↔ (𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅))
10	7, 9	bitrdi 287	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅)))
11	3, 4	elmapd 8898	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑_m 𝐵) ↔ 𝐹:𝐵⟶𝐴))
12	11	anbi1d 630	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅) ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
13	10, 12	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 ∅c0 4352 class class class wbr 5166 dom cdm 5700 Oncon0 6395 ⟶wf 6569 (class class class)co 7448 ↑_m cmap 8884 finSupp cfsupp 9431 CNF ccnf 9730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-seqom 8504 df-map 8886 df-cnf 9731

This theorem is referenced by: cantnfcl 9736 cantnfle 9740 cantnflt 9741 cantnff 9743 cantnf0 9744 cantnfrescl 9745 cantnfp1lem1 9747 cantnfp1lem2 9748 cantnfp1lem3 9749 cantnfp1 9750 oemapvali 9753 cantnflem1a 9754 cantnflem1b 9755 cantnflem1c 9756 cantnflem1d 9757 cantnflem1 9758 cantnflem3 9760 cantnf 9762 cnfcomlem 9768 cnfcom 9769 cnfcom2lem 9770 cnfcom3lem 9772 cnfcom3 9773 cantnfub 43283 cantnfresb 43286 cantnf2 43287 naddcnff 43324 naddcnffo 43326 naddcnfcom 43328 naddcnfid1 43329 naddcnfid2 43330 naddcnfass 43331

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