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Theorem cnvimadfsn 6479

Description: The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)

**Assertion**
Ref	Expression
cnvimadfsn	⊢ (^◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}

Distinct variable groups: 𝑥,𝑅,𝑦 𝑥,𝑍,𝑦

**Proof of Theorem cnvimadfsn**
Step	Hyp	Ref	Expression
1		dfima3 5127	. 2 ⊢ (^◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅)}
2		eldifvsn 3845	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍))
3	2	elv 2825	. . . . 5 ⊢ (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍)
4		vex 2824	. . . . . . 7 ⊢ 𝑦 ∈ V
5		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
6	4, 5	opelcnv 4960	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		df-br 4129	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
8	6, 7	bitr4i 187	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 ↔ 𝑥𝑅𝑦)
9	3, 8	anbi12ci 465	. . . 4 ⊢ ((𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍))
10	9	exbii 1658	. . 3 ⊢ (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍))
11	10	abbii 2354	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}
12	1, 11	eqtri 2259	1 ⊢ (^◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ≠ wne 2420 Vcvv 2821 ∖ cdif 3217 {csn 3708 ⟨cop 3711 class class class wbr 4128 ^◡ccnv 4771 “ cima 4775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-br 4129 df-opab 4191 df-xp 4778 df-cnv 4780 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785

This theorem is referenced by: (None)

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