Theorem divsfval 13626

Description: Value of the function in qusval 13621. (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 5782	. . . 4 ⊢ (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉)
3	2	a1i 9	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) → 𝐴 ∈ 𝑉))
4		19.8a 1643	. . . 4 ⊢ (𝑦 ∈ [𝐴] ∼ → ∃𝑦 𝑦 ∈ [𝐴] ∼ )
5		ecdmn0m 6841	. . . . . 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 6807	. . . . . . 7 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
9	7, 8	syl 14	. . . . . 6 ⊢ (𝜑 → dom ∼ = 𝑉)
10	9	eleq2d 2308	. . . . 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 6832	. . . . . 6 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
14		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
15		ercpbl.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
16	7	ecss 6840	. . . . . . . 8 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
17	15, 16	ssexd 4268	. . . . . . 7 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
18	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴] ∼ ∈ V)
19	1, 13, 14, 18	fvmptd3 5793	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) = [𝐴] ∼ )
20	19	eleq2d 2308	. . . 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 717	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝐴) ↔ 𝑦 ∈ [𝐴] ∼ ))
23	22	eqrdv 2236	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4187  dom cdm 4769  ‘cfv 5372   Er wer 6794  [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fv 5380  df-er 6797  df-ec 6799

This theorem is referenced by:  qusrhm  14837