Theorem cantnfs 8563

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 2622	. . . . . 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 8561	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅})
6	1, 5	syl5eq 2668	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅})
7	6	eleq2d 2687	. . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅}))
8		breq1 4656	. . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅))
9	8	elrab 3363	. . 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 7871	. . 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 384   = wceq 1483   ∈ wcel 1990  {crab 2916  ∅c0 3915   class class class wbr 4653  dom cdm 5114  Oncon0 5723  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857   finSupp cfsupp 8275   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-map 7859  df-cnf 8559

This theorem is referenced by:  cantnfcl  8564  cantnfle  8568  cantnflt  8569  cantnff  8571  cantnf0  8572  cantnfrescl  8573  cantnfp1lem1  8575  cantnfp1lem2  8576  cantnfp1lem3  8577  cantnfp1  8578  oemapvali  8581  cantnflem1a  8582  cantnflem1b  8583  cantnflem1c  8584  cantnflem1d  8585  cantnflem1  8586  cantnflem3  8588  cantnf  8590  cnfcomlem  8596  cnfcom  8597  cnfcom2lem  8598  cnfcom3lem  8600  cnfcom3  8601