Theorem tposidres 49910

Description: Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)

Hypotheses

Ref	Expression
tposidres.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposidres.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
tposidres	⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Proof of Theorem tposidres

Step	Hyp	Ref	Expression
1		ovtpos 8237	. . . 4 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (𝑋( I ↾ (𝐴 × 𝐵))𝑌)
2		df-ov 7412	. . . 4 ⊢ (𝑋( I ↾ (𝐴 × 𝐵))𝑌) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
3	1, 2	eqtri 2783	. . 3 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
4		tposidres.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		tposidres.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	4, 5	opelxpd 5687	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
7	6	fvresd 6894	. . 3 ⊢ (𝜑 → (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩) = ( I ‘⟨𝑋, 𝑌⟩))
8	3, 7	eqtrid 2807	. 2 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ( I ‘⟨𝑋, 𝑌⟩))
9		opex 5432	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
10		fvi 6950	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ V → ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩)
11	9, 10	ax-mp 5	. 2 ⊢ ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩
12	8, 11	eqtrdi 2811	1 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   I cid 5542   × cxp 5646   ↾ cres 5650  ‘cfv 6528  (class class class)co 7409  tpos ctpos 8221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-ov 7412  df-tpos 8222

This theorem is used by: (None)