Theorem eldisjlem19 38766

Description: Special case of disjlem19 38757 (together with membpartlem19 38767, this is former prtlem19 38834). (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 38663	. . . . . . . 8 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		disjlem19 38757	. . . . . . . 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 38242	. . . . . . . 8 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → [𝑢](◡ E ↾ 𝐴) = 𝑢)
7	6	eleq2d 2830	. . . . . . 7 ⊢ (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ [𝑢](◡ E ↾ 𝐴) ↔ 𝐵 ∈ 𝑢))
8	6	eqeq1d 2742	. . . . . . 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 38368	. . . . . 6 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
13	12	eceq2i 8805	. . . . 5 ⊢ [𝐵] ∼ 𝐴 = [𝐵] ≀ (◡ E ↾ 𝐴)
14	13	eqeq2i 2753	. . . 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 1537   ∈ wcel 2108   E cep 5598  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702  [cec 8761   ≀ ccoss 38135   ∼ ccoels 38136   Disj wdisjALTV 38169   ElDisj weldisj 38171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-coss 38367  df-coels 38368  df-cnvrefrel 38483  df-disjALTV 38661  df-eldisj 38663

This theorem is referenced by:  membpartlem19  38767