Theorem swapf1val 48946

Description: The object part of the swap functor. See also swapf1vala 48945. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapf2fvala.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf2fvala.b	⊢ 𝐵 = (Base‘𝑆)
swapf1val.o	⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)

Assertion

Ref	Expression
swapf1val	⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑃(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝑂(𝑥)   𝑉(𝑥)

Proof of Theorem swapf1val

Step	Hyp	Ref	Expression
1		swapf1val.o	. . 3 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
2	1	fveq2d 6908	. 2 ⊢ (𝜑 → (1st ‘(𝐶swapF𝐷)) = (1st ‘⟨𝑂, 𝑃⟩))
3		swapfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
4		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		swapf2fvala.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
6		swapf2fvala.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7	3, 4, 5, 6	swapf1vala 48945	. 2 ⊢ (𝜑 → (1st ‘(𝐶swapF𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
8	3, 4	swapfelvv 48942	. . . 4 ⊢ (𝜑 → (𝐶swapF𝐷) ∈ (V × V))
9	1, 8	eqeltrrd 2841	. . 3 ⊢ (𝜑 → ⟨𝑂, 𝑃⟩ ∈ (V × V))
10		opelxp 5719	. . . 4 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) ↔ (𝑂 ∈ V ∧ 𝑃 ∈ V))
11	10	biimpi 216	. . 3 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) → (𝑂 ∈ V ∧ 𝑃 ∈ V))
12		op1stg 8022	. . 3 ⊢ ((𝑂 ∈ V ∧ 𝑃 ∈ V) → (1st ‘⟨𝑂, 𝑃⟩) = 𝑂)
13	9, 11, 12	3syl 18	. 2 ⊢ (𝜑 → (1st ‘⟨𝑂, 𝑃⟩) = 𝑂)
14	2, 7, 13	3eqtr3rd 2785	1 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3479  {csn 4624  ⟨cop 4630  ∪ cuni 4905   ↦ cmpt 5223   × cxp 5681  ^◡ccnv 5682  ‘cfv 6559  (class class class)co 7429  1^st c1st 8008  Basecbs 17243   ×_c cxpc 18209  swap_Fcswapf 48938

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-swapf 48939

This theorem is referenced by:  swapf1a  48948  swapf1  48951  swapf1f1o  48954