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Theorem cantnfs 9354

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 2738	. . . . . 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 9352	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	eqtrid 2790	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2824	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 5073	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3617	. . 3 ⊢ (𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅} ↔ (𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅))
10	7, 9	bitrdi 286	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅)))
11	3, 4	elmapd 8587	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑_m 𝐵) ↔ 𝐹:𝐵⟶𝐴))
12	11	anbi1d 629	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝐴 ↑_m 𝐵) ∧ 𝐹 finSupp ∅) ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
13	10, 12	bitrd 278	1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 ∅c0 4253 class class class wbr 5070 dom cdm 5580 Oncon0 6251 ⟶wf 6414 (class class class)co 7255 ↑_m cmap 8573 finSupp cfsupp 9058 CNF ccnf 9349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-seqom 8249 df-map 8575 df-cnf 9350

This theorem is referenced by: cantnfcl 9355 cantnfle 9359 cantnflt 9360 cantnff 9362 cantnf0 9363 cantnfrescl 9364 cantnfp1lem1 9366 cantnfp1lem2 9367 cantnfp1lem3 9368 cantnfp1 9369 oemapvali 9372 cantnflem1a 9373 cantnflem1b 9374 cantnflem1c 9375 cantnflem1d 9376 cantnflem1 9377 cantnflem3 9379 cantnf 9381 cnfcomlem 9387 cnfcom 9388 cnfcom2lem 9389 cnfcom3lem 9391 cnfcom3 9392

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