Theorem ranrcl4lem 49617

Description: Lemma for ranrcl4 49618 and ranrcl5 49619. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypothesis

Ref	Expression
ranrcl2.l	⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)

Assertion

Ref	Expression
ranrcl4lem	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)

Proof of Theorem ranrcl4lem

Step	Hyp	Ref	Expression
1		ranrcl2.l	. . . 4 ⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)
2	1	ranrcl2 49615	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		opex 5426	. . . 4 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
5	2, 4	prcofelvv 49359	. 2 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
6		1st2nd2 8009	. 2 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
7	5, 6	syl 17	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4597   class class class wbr 5109   × cxp 5638  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969   Func cfunc 17822   −∘_F cprcof 49352   Ran cran 49585

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-prcof 49353  df-ran 49587

This theorem is referenced by:  ranrcl4  49618  ranrcl5  49619