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Theorem ssrnres 4686
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres (B ⊆ ran (𝐶A) ↔ ran (𝐶 ∩ (A × B)) = B)

Proof of Theorem ssrnres
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3131 . . . . 5 (𝐶 ∩ (A × B)) ⊆ (A × B)
2 rnss 4487 . . . . 5 ((𝐶 ∩ (A × B)) ⊆ (A × B) → ran (𝐶 ∩ (A × B)) ⊆ ran (A × B))
31, 2ax-mp 7 . . . 4 ran (𝐶 ∩ (A × B)) ⊆ ran (A × B)
4 rnxpss 4677 . . . 4 ran (A × B) ⊆ B
53, 4sstri 2927 . . 3 ran (𝐶 ∩ (A × B)) ⊆ B
6 eqss 2933 . . 3 (ran (𝐶 ∩ (A × B)) = B ↔ (ran (𝐶 ∩ (A × B)) ⊆ B B ⊆ ran (𝐶 ∩ (A × B))))
75, 6mpbiran 833 . 2 (ran (𝐶 ∩ (A × B)) = BB ⊆ ran (𝐶 ∩ (A × B)))
8 ssid 2937 . . . . . . . 8 AA
9 ssv 2938 . . . . . . . 8 B ⊆ V
10 xpss12 4368 . . . . . . . 8 ((AA B ⊆ V) → (A × B) ⊆ (A × V))
118, 9, 10mp2an 404 . . . . . . 7 (A × B) ⊆ (A × V)
12 sslin 3136 . . . . . . 7 ((A × B) ⊆ (A × V) → (𝐶 ∩ (A × B)) ⊆ (𝐶 ∩ (A × V)))
1311, 12ax-mp 7 . . . . . 6 (𝐶 ∩ (A × B)) ⊆ (𝐶 ∩ (A × V))
14 df-res 4280 . . . . . 6 (𝐶A) = (𝐶 ∩ (A × V))
1513, 14sseqtr4i 2951 . . . . 5 (𝐶 ∩ (A × B)) ⊆ (𝐶A)
16 rnss 4487 . . . . 5 ((𝐶 ∩ (A × B)) ⊆ (𝐶A) → ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A))
1715, 16ax-mp 7 . . . 4 ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A)
18 sstr 2926 . . . 4 ((B ⊆ ran (𝐶 ∩ (A × B)) ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A)) → B ⊆ ran (𝐶A))
1917, 18mpan2 403 . . 3 (B ⊆ ran (𝐶 ∩ (A × B)) → B ⊆ ran (𝐶A))
20 ssel 2912 . . . . . . 7 (B ⊆ ran (𝐶A) → (y By ran (𝐶A)))
21 vex 2534 . . . . . . . 8 y V
2221elrn2 4499 . . . . . . 7 (y ran (𝐶A) ↔ xx, y (𝐶A))
2320, 22syl6ib 150 . . . . . 6 (B ⊆ ran (𝐶A) → (y Bxx, y (𝐶A)))
2423ancrd 309 . . . . 5 (B ⊆ ran (𝐶A) → (y B → (xx, y (𝐶A) y B)))
2521elrn2 4499 . . . . . 6 (y ran (𝐶 ∩ (A × B)) ↔ xx, y (𝐶 ∩ (A × B)))
26 elin 3099 . . . . . . . 8 (⟨x, y (𝐶 ∩ (A × B)) ↔ (⟨x, y 𝐶 x, y (A × B)))
27 opelxp 4297 . . . . . . . . 9 (⟨x, y (A × B) ↔ (x A y B))
2827anbi2i 433 . . . . . . . 8 ((⟨x, y 𝐶 x, y (A × B)) ↔ (⟨x, y 𝐶 (x A y B)))
2921opelres 4540 . . . . . . . . . 10 (⟨x, y (𝐶A) ↔ (⟨x, y 𝐶 x A))
3029anbi1i 434 . . . . . . . . 9 ((⟨x, y (𝐶A) y B) ↔ ((⟨x, y 𝐶 x A) y B))
31 anass 383 . . . . . . . . 9 (((⟨x, y 𝐶 x A) y B) ↔ (⟨x, y 𝐶 (x A y B)))
3230, 31bitr2i 174 . . . . . . . 8 ((⟨x, y 𝐶 (x A y B)) ↔ (⟨x, y (𝐶A) y B))
3326, 28, 323bitri 195 . . . . . . 7 (⟨x, y (𝐶 ∩ (A × B)) ↔ (⟨x, y (𝐶A) y B))
3433exbii 1474 . . . . . 6 (xx, y (𝐶 ∩ (A × B)) ↔ x(⟨x, y (𝐶A) y B))
35 19.41v 1760 . . . . . 6 (x(⟨x, y (𝐶A) y B) ↔ (xx, y (𝐶A) y B))
3625, 34, 353bitri 195 . . . . 5 (y ran (𝐶 ∩ (A × B)) ↔ (xx, y (𝐶A) y B))
3724, 36syl6ibr 151 . . . 4 (B ⊆ ran (𝐶A) → (y By ran (𝐶 ∩ (A × B))))
3837ssrdv 2924 . . 3 (B ⊆ ran (𝐶A) → B ⊆ ran (𝐶 ∩ (A × B)))
3919, 38impbii 117 . 2 (B ⊆ ran (𝐶 ∩ (A × B)) ↔ B ⊆ ran (𝐶A))
407, 39bitr2i 174 1 (B ⊆ ran (𝐶A) ↔ ran (𝐶 ∩ (A × B)) = B)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531  cin 2889  wss 2890  cop 3349   × cxp 4266  ran crn 4269  cres 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280
This theorem is referenced by:  rninxp  4687
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