Theorem dmqsblocks 38818

Description: If the pet 38816 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 38613). It makes explicit that pet 38816 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 38613	. . 3 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))))
2		eqab2 38210	. . 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 3078	. . . 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 38370	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏)))
6	5	elv 3449	. . . . . . . . 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 1955	. . . . . . . . 9 ⊢ (∃𝑐∃𝑏(𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ (∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏))
10		excom 2163	. . . . . . . . 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 1849	. . . . . . 7 ⊢ (∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏) ↔ ∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏)))
16		19.42vv 1957	. . . . . . 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 3067	. . . 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 3066	. 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   class class class wbr 5102   E cep 5530  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633  [cec 8646   / cqs 8647   ⋉ cxrn 38141

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-oprab 7373  df-1st 7947  df-2nd 7948  df-qs 8654  df-xrn 38326

This theorem is referenced by: (None)