Theorem swapf1val 49272

Description: The object part of the swap functor. See also swapf1vala 49271. (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 → (𝐶 swapF 𝐷) = ⟨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 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
2	1	fveq2d 6830	. 2 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (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 49271	. 2 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
8	3, 4	swapfelvv 49268	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
9	1, 8	eqeltrrd 2829	. . 3 ⊢ (𝜑 → ⟨𝑂, 𝑃⟩ ∈ (V × V))
10		opelxp 5659	. . . 4 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) ↔ (𝑂 ∈ V ∧ 𝑃 ∈ V))
11	10	biimpi 216	. . 3 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) → (𝑂 ∈ V ∧ 𝑃 ∈ V))
12		op1stg 7943	. . 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 2773	1 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579  ⟨cop 4585  ∪ cuni 4861   ↦ cmpt 5176   × cxp 5621  ^◡ccnv 5622  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  Basecbs 17139   ×_c cxpc 18093   swap_F cswapf 49264

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-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-swapf 49265

This theorem is referenced by:  swapf1a  49274  swapf1  49277  swapf1f1o  49280