Theorem dmqsblocks 38891

Description: If the pet 38889 span (𝑅 ⋉ (' E | 𝐴)) partitions 𝐴, then every block 𝑢 ∈ 𝐴 is of the form [𝑣] for some 𝑣 that not only lies in the domain but also has at least one internal element 𝑐 and at least one 𝑅-target 𝑏 (cf. also the comments of qseq 38686). It makes explicit that pet 38889 gives active representatives for each block, without ever forcing 𝑣 = 𝑢. (Contributed by Peter Mazsa, 23-Nov-2025.)

Assertion

Ref	Expression
dmqsblocks	⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))

Distinct variable groups:   𝐴,𝑏,𝑐,𝑢,𝑣   𝑅,𝑏,𝑐,𝑢,𝑣

Proof of Theorem dmqsblocks

Step	Hyp	Ref	Expression
1		qseq 38686	. . 3 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))))
2		eqab2 38283	. . 3 ⊢ (∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
3	1, 2	sylbi 217	. 2 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
4		rexanid 3081	. . . 4 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))(𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)))
5		eldmxrncnvepres2 38443	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏)))
6	5	elv 3441	. . . . . . . . 9 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
7		3simpc 1150	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
8	6, 7	sylbi 217	. . . . . . . 8 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
9		exdistrv 1956	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
10		excom 2165	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
11	9, 10	bitr3i 277	. . . . . . . 8 ⊢ ((∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
12	8, 11	sylib 218	. . . . . . 7 ⊢ (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) → ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
13	12	anim1ci 616	. . . . . 6 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
14		3anass 1094	. . . . . . . 8 ⊢ ((𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
15	14	2exbii 1850	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
16		19.42vv 1958	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)) ↔ (𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
17	15, 16	sylbbr 236	. . . . . 6 ⊢ ((𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∃𝑏∃𝑐(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)) → ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
18	13, 17	syl 17	. . . . 5 ⊢ ((𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
19	18	reximi 3070	. . . 4 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))(𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))) → ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
20	4, 19	sylbir 235	. . 3 ⊢ (∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) → ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
21	20	ralimi 3069	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
22	3, 21	syl 17	1 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   class class class wbr 5086   E cep 5510  ^◡ccnv 5610  dom cdm 5611   ↾ cres 5613  [cec 8615   / cqs 8616   ⋉ cxrn 38214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-eprel 5511  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fo 6482  df-fv 6484  df-oprab 7345  df-1st 7916  df-2nd 7917  df-qs 8623  df-xrn 38399

This theorem is referenced by: (None)