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Theorem cantnfs 9626

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 2730	. . . . . 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 9624	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	eqtrid 2777	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2815	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑_m 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 5113	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3662	. . 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 8816	. . 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 1540 ∈ wcel 2109 {crab 3408 ∅c0 4299 class class class wbr 5110 dom cdm 5641 Oncon0 6335 ⟶wf 6510 (class class class)co 7390 ↑_m cmap 8802 finSupp cfsupp 9319 CNF ccnf 9621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-seqom 8419 df-map 8804 df-cnf 9622

This theorem is referenced by: cantnfcl 9627 cantnfle 9631 cantnflt 9632 cantnff 9634 cantnf0 9635 cantnfrescl 9636 cantnfp1lem1 9638 cantnfp1lem2 9639 cantnfp1lem3 9640 cantnfp1 9641 oemapvali 9644 cantnflem1a 9645 cantnflem1b 9646 cantnflem1c 9647 cantnflem1d 9648 cantnflem1 9649 cantnflem3 9651 cantnf 9653 cnfcomlem 9659 cnfcom 9660 cnfcom2lem 9661 cnfcom3lem 9663 cnfcom3 9664 cantnfub 43317 cantnfresb 43320 cantnf2 43321 naddcnff 43358 naddcnffo 43360 naddcnfcom 43362 naddcnfid1 43363 naddcnfid2 43364 naddcnfass 43365

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