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Theorem uniimaelsetpreimafv 45678
Description: The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}
Assertion
Ref Expression
uniimaelsetpreimafv ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
Distinct variable groups:   π‘₯,𝐴,𝑧   π‘₯,𝐹,𝑧   π‘₯,𝑆,𝑧   π‘₯,𝑃
Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem uniimaelsetpreimafv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}
210nelsetpreimafv 45672 . . . 4 (𝐹 Fn 𝐴 β†’ βˆ… βˆ‰ 𝑃)
3 elnelne2 3057 . . . . . 6 ((𝑆 ∈ 𝑃 ∧ βˆ… βˆ‰ 𝑃) β†’ 𝑆 β‰  βˆ…)
4 n0 4310 . . . . . 6 (𝑆 β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
53, 4sylib 217 . . . . 5 ((𝑆 ∈ 𝑃 ∧ βˆ… βˆ‰ 𝑃) β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
65expcom 415 . . . 4 (βˆ… βˆ‰ 𝑃 β†’ (𝑆 ∈ 𝑃 β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆))
72, 6syl 17 . . 3 (𝐹 Fn 𝐴 β†’ (𝑆 ∈ 𝑃 β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆))
87imp 408 . 2 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
91imaelsetpreimafv 45677 . . . . . 6 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑦 ∈ 𝑆) β†’ (𝐹 β€œ 𝑆) = {(πΉβ€˜π‘¦)})
1093expa 1119 . . . . 5 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ (𝐹 β€œ 𝑆) = {(πΉβ€˜π‘¦)})
1110unieqd 4883 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) = βˆͺ {(πΉβ€˜π‘¦)})
12 fvex 6859 . . . . 5 (πΉβ€˜π‘¦) ∈ V
1312unisn 4891 . . . 4 βˆͺ {(πΉβ€˜π‘¦)} = (πΉβ€˜π‘¦)
1411, 13eqtrdi 2789 . . 3 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) = (πΉβ€˜π‘¦))
15 dffn3 6685 . . . . . 6 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
1615biimpi 215 . . . . 5 (𝐹 Fn 𝐴 β†’ 𝐹:𝐴⟢ran 𝐹)
1716ad2antrr 725 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ 𝐹:𝐴⟢ran 𝐹)
181elsetpreimafvssdm 45668 . . . . 5 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ 𝑆 βŠ† 𝐴)
1918sselda 3948 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ 𝐴)
2017, 19ffvelcdmd 7040 . . 3 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ (πΉβ€˜π‘¦) ∈ ran 𝐹)
2114, 20eqeltrd 2834 . 2 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
228, 21exlimddv 1939 1 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2940   βˆ‰ wnel 3046  βˆƒwrex 3070  βˆ…c0 4286  {csn 4590  βˆͺ cuni 4869  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by:  imasetpreimafvbijlemf  45683  fundcmpsurbijinjpreimafv  45689
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