Theorem divsfval 13410

Description: Value of the function in qusval 13405. (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 5729	. . . 4 ⊢ (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉)
3	2	a1i 9	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉))
4		19.8a 1638	. . . 4 ⊢ (𝑦 ∈ [𝐴] ∼ → ∃𝑦 𝑦 ∈ [𝐴] ∼ )
5		ecdmn0m 6745	. . . . . 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 6711	. . . . . . 7 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
9	7, 8	syl 14	. . . . . 6 ⊢ (𝜑 → dom ∼ = 𝑉)
10	9	eleq2d 2301	. . . . 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 6736	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
14		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
15		ercpbl.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
16	7	ecss 6744	. . . . . . . 8 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
17	15, 16	ssexd 4229	. . . . . . 7 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
18	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴] ∼ ∈ V)
19	1, 13, 14, 18	fvmptd3 5740	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) = [𝐴] ∼ )
20	19	eleq2d 2301	. . . 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 712	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ ))
23	22	eqrdv 2229	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802   ↦ cmpt 4150  dom cdm 4725  ‘cfv 5326   Er wer 6698  [cec 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-er 6701  df-ec 6703

This theorem is referenced by:  qusrhm  14541