Theorem cantnfs 8601

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 2651	. . . . . 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 8599	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	syl5eq 2697	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2716	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 4688	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3396	. . 3 ⊢ (𝐹 ∈ {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅} ↔ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ∧ 𝐹 finSupp ∅))
10	7, 9	syl6bb 276	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ∧ 𝐹 finSupp ∅)))
11	3, 4	elmapd 7913	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴))
12	11	anbi1d 741	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝐴 ↑𝑚 𝐵) ∧ 𝐹 finSupp ∅) ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
13	10, 12	bitrd 268	1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {crab 2945  ∅c0 3948   class class class wbr 4685  dom cdm 5143  Oncon0 5761  ⟶wf 5922  (class class class)co 6690   ↑_𝑚 cmap 7899   finSupp cfsupp 8316   CNF ccnf 8596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-map 7901  df-cnf 8597

This theorem is referenced by:  cantnfcl  8602  cantnfle  8606  cantnflt  8607  cantnff  8609  cantnf0  8610  cantnfrescl  8611  cantnfp1lem1  8613  cantnfp1lem2  8614  cantnfp1lem3  8615  cantnfp1  8616  oemapvali  8619  cantnflem1a  8620  cantnflem1b  8621  cantnflem1c  8622  cantnflem1d  8623  cantnflem1  8624  cantnflem3  8626  cantnf  8628  cnfcomlem  8634  cnfcom  8635  cnfcom2lem  8636  cnfcom3lem  8638  cnfcom3  8639