Theorem divsfval 12971

Description: Value of the function in qusval 12966. (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 5644	. . . 4 ⊢ (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉)
3	2	a1i 9	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉))
4		19.8a 1604	. . . 4 ⊢ (𝑦 ∈ [𝐴] ∼ → ∃𝑦 𝑦 ∈ [𝐴] ∼ )
5		ecdmn0m 6636	. . . . . 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 6602	. . . . . . 7 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
9	7, 8	syl 14	. . . . . 6 ⊢ (𝜑 → dom ∼ = 𝑉)
10	9	eleq2d 2266	. . . . 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 6627	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
14		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
15		ercpbl.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
16	7	ecss 6635	. . . . . . . 8 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
17	15, 16	ssexd 4173	. . . . . . 7 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
18	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴] ∼ ∈ V)
19	1, 13, 14, 18	fvmptd3 5655	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) = [𝐴] ∼ )
20	19	eleq2d 2266	. . . 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 706	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ ))
23	22	eqrdv 2194	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763   ↦ cmpt 4094  dom cdm 4663  ‘cfv 5258   Er wer 6589  [cec 6590

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fv 5266  df-er 6592  df-ec 6594

This theorem is referenced by:  qusrhm  14084