Theorem eldisjlem19 38809

Description: Special case of disjlem19 38800 (together with membpartlem19 38810, this is former prtlem19 38878). (Contributed by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
eldisjlem19	⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉

Proof of Theorem eldisjlem19

Step	Hyp	Ref	Expression
1		df-eldisj 38706	. . . . . . . 8 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		disjlem19 38800	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ( Disj (◡ E ↾ 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
3	1, 2	biimtrid 242	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴))))
4	3	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢](◡ E ↾ 𝐴)) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
5	4	expdimp 452	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)))
6		eccnvepres3 38281	. . . . . . . 8 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = 𝑢)
7	6	eleq2d 2815	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) ↔ 𝐵 ∈ 𝑢))
8	6	eqeq1d 2732	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ([𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴) ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
9	7, 8	imbi12d 344	. . . . . 6 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → ((𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))))
10	9	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → ((𝐵 ∈ [𝑢](◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = [𝐵] ≀ (◡ E ↾ 𝐴)) ↔ (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))))
11	5, 10	mpbid 232	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴)))
12		df-coels 38410	. . . . . 6 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
13	12	eceq2i 8716	. . . . 5 ⊢ [𝐵] ∼ 𝐴 = [𝐵] ≀ (◡ E ↾ 𝐴)
14	13	eqeq2i 2743	. . . 4 ⊢ (𝑢 = [𝐵] ∼ 𝐴 ↔ 𝑢 = [𝐵] ≀ (◡ E ↾ 𝐴))
15	11, 14	imbitrrdi 252	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom (◡ E ↾ 𝐴)) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))
16	15	expimpd 453	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
17	16	ex 412	1 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   E cep 5540  ^◡ccnv 5640  dom cdm 5641   ↾ cres 5643  [cec 8672   ≀ ccoss 38176   ∼ ccoels 38177   Disj wdisjALTV 38210   ElDisj weldisj 38212

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 5254  ax-nul 5264  ax-pr 5390

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-rmo 3356  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676  df-coss 38409  df-coels 38410  df-cnvrefrel 38525  df-disjALTV 38704  df-eldisj 38706

This theorem is referenced by:  membpartlem19  38810