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Theorem cantnfs 9704

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 2735	. . . . . 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 9702	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	eqtrid 2787	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2825	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 5151	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3695	. . 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 8879	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑_m 𝐵) ↔ 𝐹:𝐵⟶𝐴))
12	11	anbi1d 631	. 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 2106 {crab 3433 ∅c0 4339 class class class wbr 5148 dom cdm 5689 Oncon0 6386 ⟶wf 6559 (class class class)co 7431 ↑_m cmap 8865 finSupp cfsupp 9399 CNF ccnf 9699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 df-map 8867 df-cnf 9700

This theorem is referenced by: cantnfcl 9705 cantnfle 9709 cantnflt 9710 cantnff 9712 cantnf0 9713 cantnfrescl 9714 cantnfp1lem1 9716 cantnfp1lem2 9717 cantnfp1lem3 9718 cantnfp1 9719 oemapvali 9722 cantnflem1a 9723 cantnflem1b 9724 cantnflem1c 9725 cantnflem1d 9726 cantnflem1 9727 cantnflem3 9729 cantnf 9731 cnfcomlem 9737 cnfcom 9738 cnfcom2lem 9739 cnfcom3lem 9741 cnfcom3 9742 cantnfub 43311 cantnfresb 43314 cantnf2 43315 naddcnff 43352 naddcnffo 43354 naddcnfcom 43356 naddcnfid1 43357 naddcnfid2 43358 naddcnfass 43359

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