Theorem orvcval2 34305

Description: Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.)

Hypotheses

Ref	Expression
orvcval.1	⊢ (𝜑 → Fun 𝑋)
orvcval.2	⊢ (𝜑 → 𝑋 ∈ 𝑉)
orvcval.3	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
orvcval2	⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})

Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋

Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem orvcval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orvcval.1	. . 3 ⊢ (𝜑 → Fun 𝑋)
2		orvcval.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		orvcval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
4	1, 2, 3	orvcval 34304	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
5		funfn 6581	. . . 4 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋)
6	1, 5	sylib 217	. . 3 ⊢ (𝜑 → 𝑋 Fn dom 𝑋)
7		fncnvima2 7066	. . 3 ⊢ (𝑋 Fn dom 𝑋 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}})
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}})
9		fvex 6906	. . . . 5 ⊢ (𝑋‘𝑧) ∈ V
10		breq1 5148	. . . . 5 ⊢ (𝑦 = (𝑋‘𝑧) → (𝑦𝑅𝐴 ↔ (𝑋‘𝑧)𝑅𝐴))
11	9, 10	elab 3665	. . . 4 ⊢ ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴)
12	11	rabbii 3425	. . 3 ⊢ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}
13	12	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})
14	4, 8, 13	3eqtrd 2770	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {cab 2703  {crab 3419   class class class wbr 5145  ^◡ccnv 5673  dom cdm 5674   “ cima 5677  Fun wfun 6540   Fn wfn 6541  ‘cfv 6546  (class class class)co 7416  ∘_RV/𝑐corvc 34302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-orvc 34303

This theorem is referenced by:  elorvc  34306