Theorem dmqsblocks 38840

Description: If the pet 38838 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 38635). It makes explicit that pet 38838 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 38635	. . 3 ⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴))))
2		eqab2 38232	. . 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 3079	. . . 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 38392	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ (𝑣 ∈ 𝐴 ∧ ∃𝑐 𝑐 ∈ 𝑣 ∧ ∃𝑏 𝑣𝑅𝑏)))
6	5	elv 3455	. . . . . . . . 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 3068	. . . 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 3067	. 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 3045  ∃wrex 3054  Vcvv 3450   class class class wbr 5109   E cep 5539  ^◡ccnv 5639  dom cdm 5640   ↾ cres 5642  [cec 8671   / cqs 8672   ⋉ cxrn 38163

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-eprel 5540  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fo 6519  df-fv 6521  df-oprab 7393  df-1st 7970  df-2nd 7971  df-qs 8679  df-xrn 38348

This theorem is referenced by: (None)