Theorem swapf1val 50045

Description: The object part of the swap functor. See also swapf1vala 50044. (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 6885	. 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 50044	. 2 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
8	3, 4	swapfelvv 50041	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
9	1, 8	eqeltrrd 2864	. . 3 ⊢ (𝜑 → ⟨𝑂, 𝑃⟩ ∈ (V × V))
10		opelxp 5697	. . . 4 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) ↔ (𝑂 ∈ V ∧ 𝑃 ∈ V))
11	10	biimpi 219	. . 3 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) → (𝑂 ∈ V ∧ 𝑃 ∈ V))
12		op1stg 7994	. . 3 ⊢ ((𝑂 ∈ V ∧ 𝑃 ∈ V) → (1st ‘⟨𝑂, 𝑃⟩) = 𝑂)
13	9, 11, 12	3syl 19	. 2 ⊢ (𝜑 → (1st ‘⟨𝑂, 𝑃⟩) = 𝑂)
14	2, 7, 13	3eqtr3rd 2807	1 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589  ⟨cop 4595  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  Basecbs 17264   ×_c cxpc 18219   swap_F cswapf 50037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-swapf 50038

This theorem is referenced by:  swapf1a  50047  swapf1  50050  swapf1f1o  50053