Theorem orvcval2 30520

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 30519	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
5		funfn 5918	. . . 4 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋)
6	1, 5	sylib 208	. . 3 ⊢ (𝜑 → 𝑋 Fn dom 𝑋)
7		fncnvima2 6339	. . 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 6201	. . . . . 6 ⊢ (𝑋‘𝑧) ∈ V
10		breq1 4656	. . . . . 6 ⊢ (𝑦 = (𝑋‘𝑧) → (𝑦𝑅𝐴 ↔ (𝑋‘𝑧)𝑅𝐴))
11	9, 10	elab 3350	. . . . 5 ⊢ ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴)
12	11	a1i 11	. . . 4 ⊢ (𝑧 ∈ dom 𝑋 → ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴))
13	12	rabbiia 3185	. . 3 ⊢ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}
14	13	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})
15	4, 8, 14	3eqtrd 2660	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916   class class class wbr 4653  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  ∘_RV/𝑐corvc 30517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-orvc 30518

This theorem is referenced by:  elorvc  30521