Theorem divsfval 17561

Description: Value of the function in qusval 17556. (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 8767	. . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉)
4	1, 3	ssexd 5294	. . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V)
5		eceq1 8758	. . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
6		ercpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
7	5, 6	fvmptg 6984	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ )
8	4, 7	sylan2 593	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ )
9	8	expcom 413	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
10	6	dmeqi 5884	. . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
11	2	ecss 8767	. . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉)
12	1, 11	ssexd 5294	. . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
13	12	ralrimivw 3136	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
14		dmmptg 6231	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉)
16	10, 15	eqtrid 2782	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉)
17	16	eleq2d 2820	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉))
18	17	notbid 318	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉))
19		ndmfv 6911	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
20	18, 19	biimtrrdi 254	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅))
21		ecdmn0 8768	. . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅)
22		erdm 8729	. . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉)
23	2, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉)
24	23	eleq2d 2820	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉))
25	24	biimpd 229	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉))
26	21, 25	biimtrrid 243	. . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉))
27	26	necon1bd 2950	. . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅))
28	20, 27	jcad 512	. . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅)))
29		eqtr3 2757	. . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ )
30	28, 29	syl6 35	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ ))
31	9, 30	pm2.61d 179	1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459  ∅c0 4308   ↦ cmpt 5201  dom cdm 5654  ‘cfv 6531   Er wer 8716  [cec 8717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-er 8719  df-ec 8721

This theorem is referenced by:  ercpbllem  17562  qusaddvallem  17565  qusgrp2  19041  frgpmhm  19746  frgpup3lem  19758  qusring2  20294  qusrhm  21237