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Theorem funsssuppss 7273
Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
Assertion
Ref Expression
funsssuppss ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Proof of Theorem funsssuppss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funss 5871 . . . . . . . . . 10 (𝐹𝐺 → (Fun 𝐺 → Fun 𝐹))
21impcom 446 . . . . . . . . 9 ((Fun 𝐺𝐹𝐺) → Fun 𝐹)
3 funfn 5882 . . . . . . . . . 10 (Fun 𝐹𝐹 Fn dom 𝐹)
43biimpi 206 . . . . . . . . 9 (Fun 𝐹𝐹 Fn dom 𝐹)
52, 4syl 17 . . . . . . . 8 ((Fun 𝐺𝐹𝐺) → 𝐹 Fn dom 𝐹)
6 funfn 5882 . . . . . . . . . 10 (Fun 𝐺𝐺 Fn dom 𝐺)
76biimpi 206 . . . . . . . . 9 (Fun 𝐺𝐺 Fn dom 𝐺)
87adantr 481 . . . . . . . 8 ((Fun 𝐺𝐹𝐺) → 𝐺 Fn dom 𝐺)
95, 8jca 554 . . . . . . 7 ((Fun 𝐺𝐹𝐺) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
1093adant3 1079 . . . . . 6 ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
1110adantr 481 . . . . 5 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
12 dmss 5288 . . . . . . . 8 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
13123ad2ant2 1081 . . . . . . 7 ((Fun 𝐺𝐹𝐺𝐺𝑉) → dom 𝐹 ⊆ dom 𝐺)
1413adantr 481 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
15 dmexg 7051 . . . . . . . 8 (𝐺𝑉 → dom 𝐺 ∈ V)
16153ad2ant3 1082 . . . . . . 7 ((Fun 𝐺𝐹𝐺𝐺𝑉) → dom 𝐺 ∈ V)
1716adantr 481 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
18 simpr 477 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
1914, 17, 183jca 1240 . . . . 5 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
2011, 19jca 554 . . . 4 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → ((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)))
21 funssfv 6171 . . . . . . . . 9 ((Fun 𝐺𝐹𝐺𝑥 ∈ dom 𝐹) → (𝐺𝑥) = (𝐹𝑥))
22213expa 1262 . . . . . . . 8 (((Fun 𝐺𝐹𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺𝑥) = (𝐹𝑥))
23 eqeq1 2625 . . . . . . . . 9 ((𝐺𝑥) = (𝐹𝑥) → ((𝐺𝑥) = 𝑍 ↔ (𝐹𝑥) = 𝑍))
2423biimpd 219 . . . . . . . 8 ((𝐺𝑥) = (𝐹𝑥) → ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2522, 24syl 17 . . . . . . 7 (((Fun 𝐺𝐹𝐺) ∧ 𝑥 ∈ dom 𝐹) → ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2625ralrimiva 2961 . . . . . 6 ((Fun 𝐺𝐹𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
27263adant3 1079 . . . . 5 ((Fun 𝐺𝐹𝐺𝐺𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2827adantr 481 . . . 4 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
29 suppfnss 7272 . . . 4 (((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)) → (∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
3020, 28, 29sylc 65 . . 3 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
3130expcom 451 . 2 (𝑍 ∈ V → ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
32 ssid 3608 . . . 4 ∅ ⊆ ∅
33 simpr 477 . . . . . . 7 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3433con3i 150 . . . . . 6 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
35 supp0prc 7250 . . . . . 6 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3634, 35syl 17 . . . . 5 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
37 simpr 477 . . . . . . 7 ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3837con3i 150 . . . . . 6 𝑍 ∈ V → ¬ (𝐺 ∈ V ∧ 𝑍 ∈ V))
39 supp0prc 7250 . . . . . 6 (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
4038, 39syl 17 . . . . 5 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
4136, 40sseq12d 3618 . . . 4 𝑍 ∈ V → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ ∅ ⊆ ∅))
4232, 41mpbiri 248 . . 3 𝑍 ∈ V → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
4342a1d 25 . 2 𝑍 ∈ V → ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
4431, 43pm2.61i 176 1 ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  wss 3559  c0 3896  dom cdm 5079  Fun wfun 5846   Fn wfn 5847  cfv 5852  (class class class)co 6610   supp csupp 7247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-supp 7248
This theorem is referenced by:  tdeglem4  23741
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