Theorem divsfval 13361

Description: Value of the function in qusval 13356. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
ercpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )

Assertion

Ref	Expression
divsfval	⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Distinct variable groups:   𝑥, ∼   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem divsfval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ercpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
2	1	mptrcl 5717	. . . 4 ⊢ (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉)
3	2	a1i 9	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉))
4		19.8a 1636	. . . 4 ⊢ (𝑦 ∈ [𝐴] ∼ → ∃𝑦 𝑦 ∈ [𝐴] ∼ )
5		ecdmn0m 6724	. . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ ∃𝑦 𝑦 ∈ [𝐴] ∼ )
6	5	biimpri 133	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ [𝐴] ∼ → 𝐴 ∈ dom ∼ )
7		ercpbl.r	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑉)
8		erdm 6690	. . . . . . 7 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
9	7, 8	syl 14	. . . . . 6 ⊢ (𝜑 → dom ∼ = 𝑉)
10	9	eleq2d 2299	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉))
11	6, 10	imbitrid 154	. . . 4 ⊢ (𝜑 → (∃𝑦 𝑦 ∈ [𝐴] ∼ → 𝐴 ∈ 𝑉))
12	4, 11	syl5 32	. . 3 ⊢ (𝜑 → (𝑦 ∈ [𝐴] ∼ → 𝐴 ∈ 𝑉))
13		eceq1 6715	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
14		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
15		ercpbl.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
16	7	ecss 6723	. . . . . . . 8 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
17	15, 16	ssexd 4224	. . . . . . 7 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
18	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴] ∼ ∈ V)
19	1, 13, 14, 18	fvmptd3 5728	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) = [𝐴] ∼ )
20	19	eleq2d 2299	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ ))
21	20	ex 115	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ )))
22	3, 12, 21	pm5.21ndd 710	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ ))
23	22	eqrdv 2227	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799   ↦ cmpt 4145  dom cdm 4719  ‘cfv 5318   Er wer 6677  [cec 6678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fv 5326  df-er 6680  df-ec 6682

This theorem is referenced by:  qusrhm  14492