Theorem divsfval 17591

Description: Value of the function in qusval 17586. (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

Step	Hyp	Ref	Expression
1		ercpbl.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
2		ercpbl.r	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑉)
3	2	ecss 8734	. . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
4	1, 3	ssexd 5285	. . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
5		eceq1 8722	. . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
6		ercpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
7	5, 6	fvmptg 6977	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ )
8	4, 7	sylan2 604	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ )
9	8	expcom 418	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
10	6	dmeqi 5885	. . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
11	2	ecss 8734	. . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉)
12	1, 11	ssexd 5285	. . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
13	12	ralrimivw 3161	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
14		dmmptg 6233	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
16	10, 15	eqtrid 2812	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉)
17	16	eleq2d 2851	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉))
18	17	notbid 321	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉))
19		ndmfv 6903	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
20	18, 19	biimtrrdi 257	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅))
21		ecdmn0 8735	. . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅)
22		erdm 8693	. . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
23	2, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉)
24	23	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉))
25	24	biimpd 232	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉))
26	21, 25	biimtrrid 246	. . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉))
27	26	necon1bd 2978	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅))
28	20, 27	jcad 521	. . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅)))
29		eqtr3 2787	. . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ )
30	28, 29	syl6 36	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
31	9, 30	pm2.61d 181	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  Vcvv 3457  ∅c0 4288   ↦ cmpt 5186  dom cdm 5652  ‘cfv 6525   Er wer 8679  [cec 8680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-er 8682  df-ec 8684

This theorem is referenced by:  ercpbllem  17592  qusaddvallem  17595  qusgrp2  19115  frgpmhm  19826  frgpup3lem  19838  qusring2  20407  qusrhm  21377