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Theorem cantnfs 9562

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 2731	. . . . . 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 9560	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	eqtrid 2778	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2817	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 5096	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3642	. . 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 8770	. . 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 1541 ∈ wcel 2111 {crab 3395 ∅c0 4282 class class class wbr 5093 dom cdm 5619 Oncon0 6312 ⟶wf 6483 (class class class)co 7352 ↑_m cmap 8756 finSupp cfsupp 9251 CNF ccnf 9557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-ov 7355 df-oprab 7356 df-mpo 7357 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-seqom 8373 df-map 8758 df-cnf 9558

This theorem is referenced by: cantnfcl 9563 cantnfle 9567 cantnflt 9568 cantnff 9570 cantnf0 9571 cantnfrescl 9572 cantnfp1lem1 9574 cantnfp1lem2 9575 cantnfp1lem3 9576 cantnfp1 9577 oemapvali 9580 cantnflem1a 9581 cantnflem1b 9582 cantnflem1c 9583 cantnflem1d 9584 cantnflem1 9585 cantnflem3 9587 cantnf 9589 cnfcomlem 9595 cnfcom 9596 cnfcom2lem 9597 cnfcom3lem 9599 cnfcom3 9600 cantnfub 43419 cantnfresb 43422 cantnf2 43423 naddcnff 43460 naddcnffo 43462 naddcnfcom 43464 naddcnfid1 43465 naddcnfid2 43466 naddcnfass 43467

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